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WORKS OF 
PROF. WALTER LORING WEBB 



PUBLISHED BY 



JOHN WILEY & SONS. 



Railroad Construction. — Theory and Practice. 

A Text-book for the Use of Students in Colleges 
and Technical Schools. Fourth Edition. Revised 
and Enlarged. 16mo. xvi -I- 777 pages and 234 
figures and plates. Morocco, $5.00. 

Problems in the Use and Adjustment of Engrlneer- 
Ins: Instruments. 

Forms for Field-notes; General Instructions for 
Extended Students' Surveys. 16mo. Morocco, 
$1.25. 

The Economics of Railroad Construction. 

Large 12mo. vii + 339 pages, 34 figures. 
$2.50. 



Cloth, 



RAILROAD CONSTRUCTION. 



THEORY AND PRACTICE. 



A TEXT-BOOK FOR THE USE OF STUDENTS 
IN COLLEGES AND TECHNICAL SCHOOLS. 



BY 



WALTER LORING WEBB, C.E., 

Jf ember American Society of Civil Engineers; Member American Railway 
Engineering and Maintenance of Way Associafion; Assistant 
Professor of Civil Engineering {Railroad En- 
gineering) in the University of 
Pennsylvania, 1893-1901; etc. 



FOURTH EDITION, REVISED AND ENLARGED. 
FIRST THOUSAND. 



NEW YORK : 

JOHN WILEY & SONS. 

London : CHAPMAIST & HALL, Limitbd. 

1908. 



-.y^^" 



UBHARY of CONliRESSl 
Iwu Gooies Hetc.v^ 

GLASS (X. AAC«JMu. 

:^l 4^ « o s 






Copyright. 1899, 1903, 1908. 

r.Y 
WALTER LORING WEBR 







PRESS Of^' 

BRAUNWORTH & CO. 

BOOKBINDERS AND PRINTERS 

BROOKLYN, N. Y. 



PREFACE TO FIRST EDITION. 



The preparation of this book was begun several years ago, . 
when much of the subject-matter treated was not to be found in 
print, or was scattered through many books and pamphlets, and 
was hence unavailable for student use. Portions of the book 
have already been printed by the mimeograph process or have 
been used as lecture-notes, and hence have been subjected to 
the refining process of class-room use. 

The author would call special attention to the following 
features : 

a. Transition curves; the multiform-compound-curve method 
is used, which has been followed by many railroads in this 
country; the particular curves here developed have the great 
advantage of being exceedingly simple, and although the method 
is not theoretically exact, it is demonstrable that the differences 
are so small that they may safely be neglected. 

h. A system of earthwork computations which makes the multi- 
plication and reduction in a single operation by means of a slide- 
rule and which enables one to compute readily the volume of the 
most complicated earthwork forms with an accuracy which is only 
limited by the precision of the cross-sectioning. 

c. The ''mass curve'' in earthwork; the theory and use of 
this very valuable process. 

d. Tables I, II, III, and IV. have been computed ah novo. 
Tables I and II were checked fafter computation) with other 
tables, which are generally considered as standard, and all 
discrepancies were further examined. They are believed to be 
perfect. 

e. Tables V, VI, VII, and IX have been borrowed, by per- 
mission, from ''Ludlow's Mathematical Tables." It is beheved 
that five-place tables give as accurate results as actual field 



IV PREFACE TO FIRST EDITION. 

practice requires. Tables VIII and X have been compiled to 
conform with Ludlow's system. 

The author wishes to acknowledge his indebtedness to Mr. 
Chas. A. Sims, civil engineer and railroad contractor, for reading 
and re\dsing the portions relating to the cost of earthwork. 

Since the book is written primarily for students of railroad 
engineering in technical institutions, the author has assumed 
the usual previous preparation in algebra, geometry, and trigo- 
nometry. 

Walter Loring Webb. 
University of Pennsylvania, 
Philadelphia, 
Jail. 1 , IWOO. 



PREFACE TO SECOND EDITION. 



Since the issue of the first edition the author has conferred 
with many noted educators in civil engineering, among them the 
late Professors E. A. Fuertes and J. B. Johnson, regarding the 
most desirable size of page for this book. The inconvenience of 
the octavo edition for field-work w^as found to be limiting its use. 
It was therefore decided to recast the whole w^ork and reduce 
the page from '' octavo '^ to ^' pocket-book '' size. Advantage 
was then taken of the opportunity to revise freely and to add 
new matter. The original text has now been almost doubled by 
the addition of several chapters on structures, train resistance, 
rolling stock, etc., and also several chapters giving the funda- 
mental principles of the economics of railroad location. Those 
who are familiar with the late Mr. Wellington's masterpiece, 
''The Economic Theory of Railway Location," will readily ap- 
preciate the author's indebtedness to that work. But w^hile the 
same general method has been followed, the author has taken 
advantage of the classification of operating expenses adopted* 
by the Interstate Commerce Commission, has used the figures* 
published by them (which were unavailable when Mr. Welling- 
ton wrote), and has developed the theory on an independent 
basis, with the exception of r. few minor details. Those who 
deny the utility of such methods of computation are referred to 
§§ 367, 426, and elsewhere for a practical discussion of that 
subject. 

The author's primary aim has been to produce a ''text-book 
for students," and the subject-matter has therefore been cut 
down to that which may properly be required of students in 

V 



VI PREFACE TO SECOND EDITION. 

the time usually allotted to railroad work in a ciAdl-engineering 
curriculum. On this account no extended discussion has been 
given to the multitudinous forms of various railroad devices 
in the chapters on structures. The aim has been to teach the 
principles and to guide the students into proper methods of 
investigation. 
January, 1903. 



PREFACE TO THIRD EDITION. 



In the present edition Tables IX and X have been entirely 
changed, the tables having been rewritten so that the values 
are given for single minutes rather than for each ten minutes. 
There has also been added a table of squares, cubes, square 
roots, cube roots, and reciprocals, which are frequently of so 
great service in computations. Advantage has also been taken 
of the opportunity to make numerous typographical and verbal 
changes. 

February, 1905. 



PREFACE TO FOURTH EDITION. 



In this edition a very extensive revision has been made in 
the chapter on Earthwork. Table XXXIII, giving the volume 
of level sections, has been added to the book, with a special 
demonstration of the method of utilizing this table for pre- 
liminary and approximate earthwork calculations. A demon- 
stration, with table, for determi^iing the economics of ties has 
also been added. In accord ari'j.c with the suggestions of Prof. 
R. B. H. Begg, of Syracuse University, additions have been 
made to Table IV, which facilitate the solution of problems in 
transition curves. Very numerous and sometimes extensive 
alterations and additions, as well as mere verbal and typo- 
graphical changes, have been made in various parts of the 
book. The chapters on Economics have been revised to make 
them conform to more recent estimates of cost of operation. 

July, 1908. 



TABLE OF CONTENTS. 



CHAPTER I. 

RAILROAD SURVEYS, 

PAGE 

Reconnoissance 1 

1. Character of a reconnoissance survey. 2. Selection of a gen- 
eral route. 3. Valley route. 4. Cross-country route. 5. Moun- 
tain route. 6, Existing maps. 7. Determination of relative 
elevations. Barometrical method. 8. Horizontal measurements, 
bearings, etc. 9. Importance of a good reconnoissance. 

Preliminary surveys 9 

10. Character of a survey. 11. Cross-section method. 12. 
Cross-sectioning. 13. Stadia method. 14. "First" and "sec- 
ond" preliminary survey. 

Location surveys 18 

15. "Paper location." 16. Surveying methods. 17. Form 
of Notes, 

CHAPTER II. 

alignment. 

Simple curves 19 

18. Designation of curves. 19. Length of a subchord. 20. 
Length of a curve. 21. Elements of a cur^^e. 22. Relation be- 
tween T, E, and J. 23. Elements of a 1° curve. 24. Exercises. 
25. Curve location by deflections. 26. Instrumental work. 27. 
Curve location by two transits. 28. Cur^'^e location by tangential 
offsets. 29. Curve location by middle ordinates. 30. Curve 
location by offsets from the long chord. 31. Use and value of the 
above methods. 32. Obstacles to location. 33. Modifications of 
location. 34. Limitations in location. 35. Determination of the 
curvature of existing track. 36. Problems. 

'OMPOUND CURVES 38 

37. Nature and use. 38. Mutual relations of the parts of a com- 
pound curve having two branches. 39. Modifications of location. 
40. Problems. 

vii 



via TABLE OF CONTENTS. 



PAGE 

Transition curves 43 

41. Superelevation of the outer rail on curves. 42. Practical 
rules for superelevation. 43. Transition from level to inclined 
track. 44. Fundamental principle of transition curves. 45. 
Multiform compound curves. 46. Required length of spiral. 47. 
To find the ordinates of a l°-per-25-feet spiral. 48. To find the 
deflections from any point of the spiral. 49. Connection of spiral 
with circular curve and with tangent. 50. Field-work. 51. To 
replace a simple curve by a curve with spirals. 52. Application 
of transition curves to compound curves. 53. To replace a com- 
pound curve by a curve with spirals. 53a. Use of Table IV. 

Vertical curves 61 

54. Necessity for their use. 55. Required length. 56. Form 
of curve. 57, Numerical example. 

CHAPTER III. 
earthwork. 

Form of excavations and embankments 60 

58. Usual form of cross-section in cut and fill. 59. Terminal 
pyramids and wedges. 60. Slopes. 61. Compound sections. 
62. Width of roadbed. 63. Form of subgrade. 64. Ditches. 
65. Effect of sodding the slopes, etc. 

Earthwork surveys 73 

66. Relation of actual volume to the numerical result. 67. 
Prismoids. 68. Cross-sectioning 69. Position of slope-stakes. 
70*. Setting slope-stakes by means of "automatic" slope-stake 
rods. 

Computation of volume 79 

71. Prismoidal formula. 72. Averaging end areas. 73. Middle 
areas. 74. Two-level ground. 75. Level sections. 76. Numeri- 
cal example, level sections. 77. Equivalent sections. 78. Equiv- 
alent level sections. 79. Three-level sections. 80. Computation 
of products. 81. Five-level sections. 82. Irregular sections. 
83. Volume of an irregular prismoid. 84. Numerical example: 
irregular sections; volume, with approximate prismoidal correc- 
tion. 85. Magnitude of the probable error of this method. S6. 
Numerical illustration of the accuracy of the approximate rule. 
87. Cross-sectioning irregular sections. 88. Side-hill work. 89. 
Borrow-pits. 90. Correction for curvature. 91. Eccentricity of 
the center of gravity. 92. Center of gravity of side-hill sections. 
93. Example of curvature correction. 94. Accuracy of earthwork 
computations. 95. Approximate computations from profiles. 

Formation of embankments 114 

96. Shrinkage of earthwork. 97. Proper allowance for shrink- 
age. 98. Methods of forming embankments. 

Computation of haul 120 

99 Nature of subject. J. 00. Mass diagram. 101. Properties 

I 



TABLE OF CONTENTS. IX 



PAGE 

of the mass curve. 102. Area of the mass curve. 103. Value of 
the mass diagram. 104. Changing the grade line. 105. Limit of 
free haul. 

Elements of the cost of earthwork 128 

106. Analysis of the total cost into items. 107. Loosening. 
108. Loading. 109. Hauling. 110. Choice of method of haul 
dependent on distance. 111. Spreading. 112. Keeping roadways 
in order. 112a. Trimming cuts to their proper cross-section. 

113. Repairs, wear depreciation, and interest on cost of plant. 

114. Superintendence and incidentals. 115. Contractor's profit 
and contingencies. 116. Limit of profitable haul. 

Blasting 149 

117. Explosives. 118. Drilling. 119. Position and direction 
of drill-holes. 120. Amount of explosive. 121. Tamping. 122. 
Exploding the charge. 123. Cost. 124. Classification of ex- 
cavated material. 125. Specifications for earthwork. 



CHAPTER IV. 



TRESTLES. 



126. Extent of use. 127. Trestles vs. embankments. 128. Two 

principal types 159 

Pile trestles If^l 

129. Pile bents. 130. Methods of driving piles. 131. Pile- 
driving formulae. 132. Pile-points and pile-shoes. 133. Details 
of design. 134. Cost of pile trestles. 
Framed trestles 167 

135. Typical design. 136. Joints. 137. Multiple-story con- 
struction. 138. Span. 139. Foundations. 140. Longitudinal 
bracing.. 141. Lateral bracing. 142. Abutments. 
Floor systems 173 

143. Stringers. 144. Corbels. 145. Guard-rails. 146 Ties on 
trestles. 147. Superelevation of the outer rail on curves. 148. 
Protection from fire. 149. Timber. 150. Cost of framed timber 
trestles. 
Design of wooden trestles 179 

151. Common practice. 152. Required elements of strength. 
153. Strength of timber. 154. Loading. 155. Factors of safety. 
156. Design of stringers. 157. Design of posts. 158. Design 
of caps and sills. 159. Bracing. 

CHAPTER V. 

tunnels. 



189 



Surveying 

160. Surface surveys. 161. Surveying down a shaft. 162, 
Underground surveys. 163. Accuracy of tunnel surveying. ' 

Design I94 

164. Cross-sections. 165. Grade. 166. Lining. 167. Shafts. 
168. Drains. 



\ 

X TABLE OF CONTENTS. 

PAGE 
Ck)NSTRUCTION , 199 

169. Headings, 170. Enlargement. 171. Distinctive features 
of various methods of construction. 172. Ventilation during con- 
struction. 173. Excavation for the portals. 174. Tunnels vs. 
open cuts. 175. Cost of tunneling. 

CHAPTER VI. 

CULVERTS AND MINOR BRIDGES. 

176. Definition and object. 177. Elements of the design 207 

Area of the waterway 208 

178. Elements involved. 179. Methods of computation of area. 
180. Empirical formulae. 181. Value of empirical formulae. 182. 
Results based on observation. 183. Degree of accuracy required. 

Pipe culverts 212 

184. Advantages. 185. Construction. 186. Iron-pipe culverts. 
187. Tile-pipe culverts. 

Box CULVERTS 216 

188. Wooden box culverts. 189. Stone box culverts. 190. Old 
rail culverts. 190a. Reinforced concrete culverts. 

Arch culverts 221 

191. Influence of design on flow. 192. Example of arch-cul- 
vert design. 

Minor openings 222 

193. Cattle-guards. 194. Cattle-passes. 195. Standard stringer 
and I-beam bridges. 

CHAPTER VII. 



196. Purpose and requirements. 197. Materials. 198. Cross- 
sections. 199. Methods of laying ballast. 200. Cost 227 

CHAPTER VIII. 

ties and other forms of rail support. 

201. Various methods of supporting rails. 202. Economics of 

ties 237 

Wooden ties 238 

203. Choice of wood. 204. Durability. 205. Dimensions. 
206. Spacing. 207. Specifications. 208. Regulations for laying 
and renewing ties. 209, Cost of ties. 

Preservative processes for wooden ties 242 

210. General principle. 211. Vulcanizing. 212. Creosoting, 
213. Burnettizing. 214. Kyanizing. 215. Wellhouse (or zinc- 
tannin) process. 216. Cost of treating. 217, Economics of 
treated ties. 



TABLE OF CONTENTS. XI 



?AGE 

Metal ties 250 

218. Extent of use. 219. Durability. 220. Form and dimen- 
sions of metal cross-ties. 221. Fastenings. 222. Cost. 223. 
Bowls or plates. 224. Longitudinals. 224a. Reinforced concrete 



ties. 



CHAPTER IX. 



225. Early forms. 226. Present standard forms. 227. Weight 
for various kinds of traffic. 228. Effect of stiffness on traction. 
229. Length of rails. 230. Expansion of rails. 231. Rules for 
allowing for temperature. 232. Chemical composition. 233. 
Testing. 233a. Proposed standard specifications f or . steel rails. 
234. Rail wear on tangents. 235. Rail wear on curves. 236. 
Cost of rails 256 

CHAPTER X. 

rail-fastenings. 

Rail-joints 270 

237. Theoretical requirements for a perfect joint. 238. Effi- 
ciency of the ordinary angle-bar. 239. Effect of rail-gap at joints. 
240. Supported, suspended, and bridge joints. 241. Failures of 
rail-joints. 242. Standard angle-bars. 243. Later designs of rail- 
joints. 243a. Proposed specifications for steel splice-bars. 

Tie-plates : 276 

244. Advantages. 245. Elements of the design. 246. Methods, 
of setting. 

Spikes 280 

247. Requirements. 248. Driving. 249. Screws and bolts. 
250. Wooden spikes. \ 

Track-bolts and nut-locks 284 

251. Essential requirements. 252. Design of track-bolts. 253. 
Design of nut-locks. 

CHAPTER XI. 

switches and crossings. 

Switch constrtjction 289 

254. Essential elements of a switch. 255. Frogs. 256. To find 
the frog number. 257. Stub switches. 258. Point switches. 259. 
Switch-stands. 260. Tie-rods. 261. Guard-rails. 

Mathematical design of switches 297 

262. Design with circular lead rails. 263. Effect of straight frog- 
rails. 264. Effect of straight point-rails. 265. Combined effect 
of straight frog-rails and straight point-rails. 266. Comparison of 
the above methods. 267. Dimensions for a turnout from the 
outer side of a curved track. 268. Dimensions for a turnout from 
the INNER side of a curved track. 269. Double turnout from a 



:^1 TABLE OF CONTENTS 



PAGE 

straight track. 270. Two turnouts on the same siJe. 271. Con- 
necting curve from a straight track. 272. -Connecting curve from 
a curved track to the outside. 273. Connecting; curve from a 
curved track to the inside. 274. Crossover between two parallel 
straight tracks. 275. Crossover between two parallel curved 
tracks. 276. Practical rules for switch-laying. 

Crossings 319 

277. Two straight tracks. 278. One straight and one curved 
track. 279. Two curved tracks. 279a. Slips. 

CHAPTER XII. 
miscellaneous structures and buildings. 

Water stations and water supply 320 

280. Location. 281. Required qualities of water. 282. Tanks. 
283. Pumping. 284. Track tanks. 285. Stand pipes. 

Buildings 331 

286. Station platforms. 287. Minor stations. 288. Section 
houses. 289. Engine houses. 

Snow structures 336 

290. Snow fences. 291. Snow sheds. 
Turntables 33S 

CHAPTER XIII. 
yards and terminals. 

293. Value of proper design. 294. Divisions of the subject ... . 340 

Freight yards 341 

295. General principles. 296. Relation of yard to main tracks. 
297. Minor freight yards. 298. Transfer cranes. 299. Track 
scales. 
Engine yards 348 

300. General principles. 

Passenger terminals ••• 350 

CHAPTER XIV. 
block signaling. 

General principles 351 

301. Two fundamental systems. 302. Manual systems. 303. 
Development of the manual system. 304. Permissive blocking. 
305. Automatic systems. 306. Distant signals. 307. Advance 
signals. 

Mechanical details 357 

308. Signals. 309. Wires and pipes. 310. Track circuit for 
automatic signaling. 



TABLE OF CONTENTS. XIU 

CHAPTER XV. 

ROLLING STOCK. 

PAGE 

Wheels and rails 363 

311. Effect of rigidly attaching wheels to their axles. 312. 
Effect of parallel axles. 313. Effect of coning wheels. 314. 
Effect of flanging locomotive driving wheels. 315. Action of a 
locomotive pilot-truck. 

locomotives. 

General structure 370 

316. Frame. 317. Boiler. 318. Fire box. 319. Coal con- 
sumption. 320. Heating surface. 321. Loss of efficiency of 
steam pressure. 322. Tractive power. 

Running gear 381 

323. Tyr 23 of running gear. 324. Equalizing levers. 325. 
Counterbalj-ncing, 326. Mutual relations of the boiler power, 
tractive power and cylinder power for various ^types. 327. Life 
of locomotives. 

CARS. 

328. Capacity and size of cars. 329. Stresses to which car- 
frames are subjected. 330. The use of metal. 331. Draft gear. 
332. Gauge of wheels and form of wheel tread 893 

train-brakes. 

333. Introduction. 334. Laws of friction as applied to this 

problem 399 

VIechanism of brakes 403 

335. Hand-brakes. 336. "Straight" air brakes. 337. Auto- 
matic air brakes. 338. Tests to measure the efficiency of brakes. 
339. Brake shoes. 

CHAPTER XVI. 

train resistance. 

340. Classification of the various forms. 341. Resistances inter- 
nal to the locomotive. 342. Velocity resistances. 343. Wheel 
resistances. 344. Grade resistance. 345. Curve resistance. 346. 
Brake resistances. 347. Inertia resistance. 348. Dynamometer 
tests. 349. Gravity or "drop" tests. 350. Formulae for train 
resistance 409 

CHAPTER XVII. 
cost of railroads. 

351. General considerations. 352. Preliminary financiering. 
353. Surv^eys and engineering expenses. 354. Land and land 



XIV TABLE OF CONTENTS 



PAGE 

damages. 355. Clearing and grubbing. 356. Earthwork. 357. 
Bridges, trestles and culverts. 358. Trackwork. 359. Buildings 
and miscellaneous structures. 360. Interest on construction. 
361. Telegraph lines. 362. Detailed estimate of the cost of aline 
of road 426 



PART II. 
RAILROAD ECONOMICS. 

CHAPTER XVIII. 

INTRODUCTION. 

363. The magnitude of railroad business. 364. Cost of trans- 
portation. 365. Study of railroad economics — its nature and 
limitations. 366. Outline of the engineer's duties. 367. Justi- 
fication of such methods of computation 436 

CHAPTER XIX. 

THE PROMOTION OF RAILROAD PROJECTS. 

368. Method of formation of railroad corporations. 369. The 
two classes of financial interests, the security and profits of each. 

370. The small margin between profit and loss to the projectors. 

371. Extent to which a railroad is a monopoly. 372. Profit 
resulting from an increase in business done; loss resulting from a 
decrease. 373. Estimation of probable volume of traffic, and of 
probable growth. 374. Probable number of trains per day. In- 
crease with growth of traffic. 375. Effect on traffic of an increase 
in facilities. 376. Loss caused by inconvenient terminals and 
by stations far removed from business centres, 377. General 
principles which should govern the expenditure of money for 
railroad purposes 441 

CHAPTER XX. 

OPERATING EXPENSES. 

378. Distribution of gross revenue. 379. Fourfold distribution 
of operating expenses. 380. Operating expenses per train mile. 
381. Reasons for uniformity in expenses per train mile. 382. 
Detailed classification of expenses with ratios to the total expense. 
383. Elements of the cost (with variations and tendencies) of the 
various items 454 



TABLE OF CONTENTS. XV 



PAGE 

Maintenance of way 463 

384. Item 1. Repairs of roadway. 385. Item 2. Renewal of 
rails. 386. Item 3. Renewal of ties. 387. Item 4. Repairs 
and renewals of bridges and culverts. 388. Items 5 to 10. Re- 
pairs and renewals of fences, road crossings, and cattle guards — 
of buildings and fixtures — of docks and wharves — of telegraph 
plant; stationery and printing; and "other expenses." 

Maintenance of equipment 466 

389. Item 11. Superintendence. 390. Item 12. Repairs and 
renewals of locomotives. 391. Items 13, 14, and 15. Repairs and 
renewals of passenger cars, of freight cars, and of work cars. 392. 
Items 16, 17, 18, and 19. Repairs and renewals of marine equip- 
ment, of shop machinery and tools; stationery and printing ; other 
expenses. 

Conducting transportation 467 

393. Item 20. Superintendence. 394. Item 21. Engine and 
roundhouse men. 395. Item 22. Fuel for locomotives. 396. 
Items 23, 24, and 25. Water supply; oil, tallow, and waste; other 
supplies for locomotives. 397. Item 26. Train service. 398. 
Item 27. Train supplies and expenses. 399. Items 28, 29, 30, and 
31. Switchmen, flagmen, and watchmen, telegraph expenses; 
station service; and station supplies. 400. Items 32, 33, and 34. 
Switching service — balance; car mileage — balance; hire of equip- 
ment. 401. Items 35, 36, and 37. Loss and damage; injuries to 
persons; clearing wrecks. 402. Items 38 to 53. 



CHAPTER XXI. 
distancb. 

403. Relation of distance to rates and •xpenses. 404. The 
conditions other than distance that affect the cost; reasons why 

rates are usually based on distance 471 

Effect of distance on operating expenses 472 

405. Effect of slight changes in distance on maintenance of way. 
406. Effect on maintenance of equipment. 407. Effect on con- 
ducting transportation. 408. Estimate of total effect on expenses 
of small changes in distance (measured in feet) ; estimate for dis- 
tances measured in miles. 
Effect of distance on receipts 479 

409. Classification of traffic. 410. Method of division of through 
rates between the roads run over. 411. Effect of a change in the 
length of the home road on its receipts from through competitive 
traffic. 412. The most advantageous conditions for roads forming 
part of a through competitive route. 413. Effect of the variations 
in the length of haul and the classes of the business actually done. 
414. General conclusions regarding a change in distance. 415. 
Justification of decreasing distance to save time. 416. Effect of 
change of distance on the business done. 



XVI TABLE OF CONTENTS. 



CHAPTER XXII. 

CURVATURE. 

PAGE 

417. General objections to curvature, 418. Financial value of 
the danger of accident due to curvature. 419. Effect of curvature 

on travel. 420. Effect on operation of trains 484 

Effect of curvature on operating expenses 488 

421. Relation of radius of curvature and of degrees of central 
angle to operating expenses. 422. Effect of curvature on mainte- 
nance of way. 423. Effect of curvature on maintenance of equip- 
ment. 424. Effect of curvature on conducting transportation, 
425. Estimate of total effect per degree of central angle. 426. 
Reliability and value of the above estimate. 

Compensation for curvature 494 

427. Reasons for compensation. 428. The proper rate of com- 
pensation. 429. The limitations of maximum curvature. 



CHAPTER XXIII. 
grade. 

430. Two distinct effects of grade. 431. Application to the 
movement of trains of the laws of accelerated motion. 432. Con- 
struction of a virtual profile. 433. Use value and possible misuse. 
434. Undulatory grades; advantages, disadvantages, and safe 
limits 500 

>MlNOR GRADES 507 

435. Basis of cost of minor grades. 436. Classification of minor 
grades. 437. Effect on operating expenses. 438. Estimate of 
the cost of one foot of change of elevation. 439. Operating value 
of the filling of a sag in a grade. 

Ruling grades 514 

440. Definition. 441. Choice of ruling grades. 442. Maximum 
train load on any grade, 443. Proportion of traffic affected by 
the ruling grade. 444. Financial value of increasing the train 
load. 445. Operating value of a reduction in the rate of the ruling 
grade. 

Pusher grades 522 

446. General principles underlying the use of pusher engines. 
447. Balance of grades for pusher service. 448. Operation of 
pusher engines. 449. Length of a pusher grade. 450. Cost of 
pusher engine service. 451. Numerical comparison of pusher and 
through grades. 

Balance of grades for unequal traffic 530 

452. Nature of the subject. 453. Computation of the theoreti- 
cal balance. 454. Computation of relative traffic. 



TABLE OF CONTENTS. XVll 

CHAPTER XXIV. 

THE IMPROVEMENT OF OLD LINES. 

FACE 

455. Classification of improvements. 456. Advantages of re- 
locations. 457. Disadvantages of re-locations 534 

Reduction of virtual grade 527 

458. Obtaining data for computations. 459. Use of the data 
obtained. 460. Reducing the starting grade at stations. 



Appendix. The ad.iustments of instruments 542 

Tables. 

I. Radii of curves 552 

II. Tangents, external distances, and long chords for a 1° curve 556 

III. Switch leads and distances 559 

IV. Transition curves 560 

V. Logarithms of numbers 869 

VI. Logarithmic sines and tangents of small angles 889 

VII. Logarithmic sines, cosines, tangents, and cotangents 892 

VIII. Logarithmic versed sines and external secants G37 

IX. Natural sines, cosines, tangents and cotangents 683 

X. Natural versed sines and external secants 706 

XL Reduction of barometer reading to 32° F 729 

XII. Barometric elevations 730 

XIII. Coefficients for corrections for temperature and humidity. . . 729 

XIV. Capacity of cylindrical water-tanks in United States standard 

gallons of 231 cubic inches 329 

XV. Number of cross ties per mile 430 

XVI. Tons per mile (with cost) of rails of various weights 431 

XVII. Splice bars and bolts per mile of track 432 

XVIII. Railroad spikes 433 

XIX. Track bolts 433 

XX. Classification of operating expenses of all railroads 460-463 

XXI. Effect on operating expenses of changes in distance 478 

XXII. Effect on operating expenses of changes in curvature 492 

XXIII. Velocity head of trains 503 

XXIV. Effect on operating expenses of change.<5 in grade 511 

XXV. Tractive power of locomotives 516 

XXVI. Total train resistance per ton on various grades 518 

XXVII, Cost of an additional train to handle a given traffic 521 

XXVIII. Balanced grades for one, two and three engines 526 

XXIX. Cost per mile of a pusher engine 529 

XXX. Useful trigonometrical formulse 731 

XXXI. Useful formulse and constants 733 

f XXXII, Squares, cubes, square roots, cube roots and reciprocals .... 734 

XXXIII. Cubic yards per 100 feet of level sections 751 

XXXIV. Annual charge against a tie, based on the original cost and as- 

sumed life of the tie 754 

Index 755 



EAILROAD CONSTRUCTION. 



CHAPTER I. 

RAILROAD SURVEYS. 

The proper conduct of railroad surveys presupposes an 
adequate knowledge of almost the whole subject of railioad 
engineering, and particularly of some of the complicated ques- 
tions of Railroad Economics, which are not generally studied 
except at the latter part of a course in railroad engineering, if 
at all. This chapter will therefore be chiefly devoted to methods 
of instrumental work, and the problem of choosing a general 
route will be considered only as it is influenced by the topog- 
raphy or by the application of those elementary principles of 
Railroad Economics which are self-evident or which may be 
accepted by the student until he has had an opportunity of 
studying those principles in detail 

RECONNOrSSANCE SURVEYS. 

1. Character of a reconnoissance survey. A reconnoissance 
survey is a very hasty examination of a belt of country to de- 
termine which of all possible or suggested routes is the most 
promising and best worthy of a more detailed survey. It is 
essentially very rough and rapid. It aims to discover those 
salient features which instantly stamp one route as distinctly 
superior to another and so narrow the choice to routes which 
are so nearly equal in value that a more detailed survey is nec- 
essary to decide between them. 

2. Selection of a general route. The general question of 
running a railroad between two towns is usually a financial rather 



2 RAILROAD CONSTRUCTION. * § 3. 

than an engineering question. Financial considerations usually 
determine that a road must pass through certain more or less 
important towns between its termini. When a railroad runs 
through a thickly settled and very flat country, where, from a 
topographical standpoint, the road may be run by any desired 
route, the ^'right-of-way agent'' sometimes has a greater influ- 
ence in locating the road than the engineer. But such modifi- 
cations of alignment, on account of business considerations, are 
foreign to the engineer's side of the subject, and it will be here- 
after assumed that topography alone determines the location of 
the line: The consideration of those larger questions combin- 
ing finance and engineering (such as passing by a town on ac- 
count of the necessar^^ introduction of heavy grades in order to 
reach it) will be taken up in Chap. XIX, et seq. 

3. Valley route. This is perhaps the simplest problem. If 
the two to^\^ls to be connected lie in the same valley, it is fre- 
quently only necessary to run a line which shall have a nearly 
uniform grade. The reconnoissance problem consists largely in 
determining the difference of elevation of the two termini of 
this division and the approximate horizontal distance so that the 
proper grade may be chosen. If there is a large river running 
through the valley, the road ^\'ill probably remain on one side 
or the other throughout the whole distance, and both banks 
should be examined by the reconnoissance party to determine 
which is preferable. If the river may be easily bridged, both 
banks may be alternately used, especially when better alignment 
is thereby secured. A river valley has usually a steeper slope 
in the upper part than in the lower part. A uniform grade 
throughout the valley will therefore require that the road climbs 
up the side slopes in the lower part of the valley. In case the 
''ruling grade" * for the w^hole road is as great as or greater 
than the steepest natural valley slope, more freedom may be 
used in adopting that alignment which has the least cost — 
regardless of grade. The natural slope of large rivers is almost 
invariably so low that grade has no influence in determining the 
choice of location. When bridging is necessary, the river 
banks should be examined for suitable locations for abutments 



* The ruling grade may here be loosely defined as the maximum grade 
which is permissible. This definition is not strictly true, as may be seen later 
when studying Railroad Economics, but it may here serve the purpose. 



§ 4. RAILROAD SURVEYS. 3 

and piers. If the soil is soft and treacherous, much difficulty 
may be experienced and the choice of route may be largely 
determined by the difficulty of bridging the river except at 
certain favorable places. 

4. Cross-country route. A cross-country route always has one 
or more summits to be crossed. The problem becomes more 
complex on account of the greater number of possible solutions 
and the difficulty of properly weighing the advantages and dis- 
advantages of each. The general aim should be to choose the 
lowest summits and the highest stream crossings, provided that 
by so doing the grades between these determining points shall 
be as low as possible and shall not be greater than the ruling 
grade of the road. Nearly all railroads combine cross-country 
and valley routes to some extent. Usually the steepest natural 
slopes are to be found on the cross-country routes, and also the 
greatest difficulty in securing a low through grade. An approx- 
imate determination of the ruling grade is usually made during 
the reconnoissance. If the ruling grade has been previously 
decided on by other considerations, the leading feature of the 
reconnoissance survey will be the determination of a general 
route along which it will be possible to survey a line whose 
maximum grade shall not exceed the ruling grade. 

5. Mountain route. The streams of a mountainous region 
frequently have a slope exceeding the desired ruling grade. In 
such cases there is no possibility of securing the desired grade 
by following the streams. The penetration of such a region 
may only be accomplished by '' development '^ — accompanied 
perhaps by tunneling. '' Development " consists in deliber- 
ately increasing the length of the road between two extremes 
of elevation so that the rate of grade shall be as low as desired. 
The usual method of accomplishing this is to take advantage of 
some convenient formation of the ground to introduce some 
lateral deviation. The methods may be somewhat classified as 
follows : 

(a) Running the line up a convenient lateral valley, turning 
a sharp curve and working back up the opposite slope. As 
shown in Fig. 1, the considerable rise between A and B was 
surmounted by starting off in a very different direction from 
the general direction of the road; then, when about one-half of 
the desired rise had been obtained, the line crossed the valley 
aad continued the climb along the opposite slope, (b) Switch- 



RAILROAD COXSTRUCTIOX. 



§5. 



back. On the steep side-hill BCD (Fig. 1) a very considerable 
gain in elevation was accomplished by the switchback CD, 
The gain in elevation from B to D is very great. On the other 
hand, the speed must always be slow; there are two complete 
stoppages of the train for each run; all trains must run back- 
ward from C to D. (c) Bridge spiral. When a valley is so 
narrow at some point that a bridge or viaduct of reasonable 
length can span the valley at a considerable elevation above ttiQ 




Fig. 1. 

bottom of the valley, a bridge spiral may be desirable. In Fig. 2 
the line ascends the stream valley past A, crosses the stream at 
B, works back to the narrow place at C, and there crosses itself, 
haAing gained perhaps 100 feet in elevation, (d) Tunnel 
spiral (Fig. 3). This is the reverse of the previous plan. It 
implies a thin steep ridge, so thin at some place that a tunne) 
through it vnll not be excessively long. Switchbacks and 
spirals are sometimes necessar}^ in mountainous countries, but 
they should not be considered as normal types of construction. 
A region must be very difficult if these de\^ces cannot be avoided. 
On Plate I are sho^n three separate ways (as actually con- 
structed) of running a railroad between two points a little over 
three miles apart and having a difference of elevation of nearly 



§5. 



RAILROAD SURVEYS. 



1100 feet. At A the Central R. R. of New Jersey runs under 
the Lehigh Valley R. R. and soon turns off to the northeast for 
about six miles, then doubles back, reaching jD, a fall of about 
1050 feet with a track distance of about 12.7 miles. The 
L. V. R. R. at A runs to the westward for six to seven miles, 





Fig. 2. 



Fig. 3. 



then turns back until the roads are again close together at D. 
The track distance is about 14 miles and the drop a little greater, 
since at A the L. V. R. R. crosses over the other, while at D they 
are at practically the same level. From B to C the distance is 
over eleven miles. From A directly down to D the C. R. R. of 
N. J. runs a '' gravity" road, used exclusively for freight, on 
which cars alone are hauled by cable. The main-line routes 
are remarkable examples of sheer "development.'' Even as 
constructed the L. V. R. R. has a grade of about 95 feet per 
mile, and this grade has proved so excessive for freight work 
that the company has constructed a cut-off (not shown on the 
map) which leaves the main line at A, nearly parallels the 
C. R. R. to C, and then running in a northeasterly direction 
again joins the main line beyond Wilkesbarre. The grade is 
thereby cut down to 65 feet per mile. 

Rack railways and cable roads, although types of mountain 
railroad construction, will not be here considered. 



RAILROAD CONSTRUCTION. 



§6. 



6. Existing maps. The maps of the U. S. Geological Survey- 
are exceedingly valuable as far as they have been completed. 
So far as topographical considerations are concerned, they 
almost dispense with the necessity for the reconnoissance and 
'^ first preliminary" surveys. Some of the State Survey maps 
will give practically the same information. County and town- 
ship maps can often be used for considerable information as to the 
relative horizontal position of governing points, and even some 
approximate data regarding elevations may be obtained by a 
study of the streams. Of course such information will not dis- 
pense with surveys, but will assist in so planning them as to 
obtain the best information with the least work. When the 
relative horizontal positions of points are reliably indicated on a 
map, the reconnoissance may be reduced to the determination 
of the relative elevations of the governing points of the route. 

7. Determination of relative elevations. A recent description 
of European methods includes spirit-leveling in the reconnois- 
sance work. This may be due to the fact that, as indicated 
above, previous topographical surveys have rendered unnecessary 
the ''exploratory" survey which is required in a new country, 
and that their reconnoissance really corresponds more nearly to 
our preliminary. 

The perfection to which barometrical methods have been 
brought has rendered it possible to determine differences of 
elevation with sufficient accuracy for reconnoissance purposes 
by the combined use of a mercurial and an aneroid barometer. 
The mercurial barometer should be kept at "headquarters," and 
readings should be taken on it at such frequent intervals that 
any fluctuation is noted, and throughout the period that observa- 
tions with the aneroid are taken in the field. At each observa- 
tion there should also be recorded the time, the reading of the 
attached thermometer, and the temperature of the external 
air. For uniformity, the mercurial readings should then be 
*' reduced to 32° F." The form of notes for the mercurial 
barometer readings should be as follows : 



Time. 


Merc. 
Barom. 


Attached 
Therm. 


Reduction 
to 32° F. 


External 
Therm. 


Corrected 
reading. 


7:00 A.M. 
:15 
:30 
:45 


29.872 

.866 
.858 
.850 


72° 
73.5 
75 
76 


— .117 
.121 
.125. 
.127 


73° 
75 

76 

77 


29.755 
.745 
.733 
.723 



§ 7. RAILROAD SURVEYS. • 7 

The corrections in column 4 are derived from Table XI by 
interpolatioQ. 

Before starting out, a reading of the aneroid should be taken 
at headquarters coincident with a reading of the mercurial. 
The difference is one value of the correction to the aneroid. 
As soon as the aneroid is brought back another comparison of 
readings should be made. Even though there has been con- 
siderable rise or fall of pressure in the interval, the difference 
in readings (the correction) should be substantially the same 
provided the aneroid is a good instrument. If the difference 
of elevation is excessive (as when climbing a high mountain) 
even the best aneroid will "lag'' and not recover its normal 
reading for several hours, but this does not apply to such dif- 
ferences of elevation as are met with in railroad work. The 
best aneroids read directly to yj^ of an inch of mercury and 
may be estimated to yw^ ^^ ^^ inoh — which corresponds 
to about 0.9 foot difference of elevation. In the field there 
should be read, at each point whose elevation is desired, the 
aneroid, the time, and the temperature. These readings, cor- 
rected by the mean value of the correction between the aneroid 
and the mercurial, should then be combined with the reading 
of the mercurial (interpolated if necessary) for the times of 
the aneroid observations and the difference of elevation ob- 
tained. The field notes for the aneroid should be taken as 
shown in the first four columns of the tabular form. The " cor- 
rected aneroid'' readings of column 5 are found by correcting 
the readings of column 3 by the mean difference between the 
mercurial and aneroid when compared at morning and night. 
Column 6 is a copy of the "corrected readings" from the office 
notes, interpolated when necessary for the proper time. Column 
7 is similarly obtained. Col. 8 is obtained from cols. 4 and 5, 
and col. 9 from cols. 6 and 7, with the aid of Table XII. The 
correction for temperature (col. 11), which is generally small 
unless the difference of elevation is large, is obtained with the 
aid of Table XIII. The elevations in Table XII are elevations 
above an assumed datum plane, where under the given atmos- 
pheric conditions the mercurial reading would be 30''. Of 
course the position of this assumed plane changes wdth varying 
atmospheric conditions and so the elevations are to be con- 
sidered as relative and their difference taken. [See the author's 
"Problems in the Use and Adjustment of Engineering In- 



RAILROAD CONSTRUCTION. 



§8. 



(Left-hand page of Notes.) 



Time. 


Place. 


Aneroid. 


Therm. 


Corr. 
Aner. 


Corr. 
Merc. 


7:00 


Office 
JO 
saddle-back 
river cross. 


29.628 
29.662 
29.374 
29.548 


73° 

72° 
63° 

70° 




29.755 


7:10 
7:30 
7:50 


29.789 
29.501 
29.675 


29.748 
29.733 
29.720 



struments," Prob. 22.] Important points should be observed 
more than once if possible. Such duplicate observations will be 
found to give surprisingly concordant results even when a 
general fluctuation of atmospheric pressure so modifies the 
tabulated readings that an agreement is not at first apparent. 
Variations of pressure produced by high winds, thunder-storms, 
etc., vnll generally vitiate possible accuracy by this method. 
By "headquarters" is meant any place whose elevation above 
any given datum is knowTi and where the mercurial may be 
placed and observed while observations within a range of several 
miles are made with the aneroid. If necessary, the elevation of 
a new headquarters may be determined by the above method, 
but there should be if possible several independent observations 
whose accordance ^vill give a fair idea of their accuracy. 

The above method should be neither slighted nor used for 
more than it is worth. When properly used, the errors are 
compensating rather than cumulative. ^Mien used, for example, 
to determine that a pass B is 260 feet higher than a determined 
bridge crossing at A which is six miles distant, and that another 
pass C is 310 feet higher than A and is ten miles distant, the 
figures, even with all necessary allowances for inaccuracy, A\dll 
give an engineer a good idea as to the choice of route especially 
as affected by ruling grade. There is no comparison between 
the time and labor involved in obtaining the above information 
by barometric and by spirit-leveling methods, and for recon- 
noissance purposes the added accuracy of the spirit-leveling 
method is hardly worth its cost. 

8. Horizontal measurements, bearings, etc. When there is 
no map which may be depended on, or when only a skeleton 
map is obtainable, a rapid survey, sufficiently accurate for the 
purpose, may be made by using a pocket compass for bearings 
and a telemeter, odometer, or pedometer for distances. The 
telemeter [stadia] is more accurate, but it requires a definite clear 



§9. 



RAILROAD SURVEYS. 



(Right-hand page of Notes.) 



Temp, at 
headqu. 


Approx. 
field read. 


Approx. 
headq. read. 


Diff. 


Corr. for 
temp. 


Diff. 
elev. 


75** 
76 

77 


192 
457 
297 


230 
244 
256 


- 38 
+ 213 
+ 41 


-(+ 2) 
+ ( + 10) 
+ (+ 2) 


— 40 
+ 223 
+ 43 



sight from station to station, which may be difficult through a 
wooded country. The odometer, which records the revolutions 
of a wheel of known circumference, may be used even in rough 
and wooded country, and the results may be depended on to a 
small percentage. The pedometer (or pace-measurer) depends 
for its accuracy on the actual movement of the mechanism for 
each pace and on the uniformity of the pacing. Its results are 
necessarily rough and approximate, but it may be used to fill 
in some intermediate points in a large skeleton map. A hand- 
level is also useful in determining the relative elevation of various 
topographical features which may have some bearing on the 
proper location of the road. 

9. Importance of a good reconnoissance. The foregoing in- 
struments and methods should be considered only as aids in 
exercising an educated common sense, without which a proper 
location cannot be made. The reconnoissance survey should 
command the best talent and the greatest experience available. 
If the general route is properly chosen, a comparatively low 
order of engineering skill can fill in a location which will prove 
a paying railroad property ; but if the general route is so chosen 
that the ruling grades are high and the business obtained is small 
and subject to competition, no amount of perfection in detailed 
alignment or roadbed construction can make the road a profitable 
investment. 



PRELIMINARY SUR^'^EYS. 

10. Character of survey. A preliminary railroad survey is 
properly a topographical survey of a belt of country which has 
been selected during the reconnoissance and within which it is 
estimated that the located line will lie. The width of this belt 
will depend on the character of the country. When a railroad 
is to follow a river having very steep banks the choice of loca- 
tion is sometimes limited at places to a very few feet of width 



10 



RAILROAD CONSTRUCTION. 



§ 11 



and the belt to be surveyed may be correspondingly narrowed. 
In very flat country the desired width may be only limited by the 
ability to survey points with sufficient accuracy at a considerable 
distance from what may be called the '^backbone line" of the 
survey. 

II. Cross-section method. This is the only feasible method 
in a wooded country, and is employed by many for all kinds 
of country. The backbone line is surveyed either by observ- 
ing magnetic bearings with a compass or by carrying forward 




Fig. 4. 



absolute azimuths with a transit. The compass method nas 
the disadvantages of limited accuracy and the possibility of 
considerable local error owing to local attraction. On the other 
hand there are the advantages of greater simplicity, no necessity 



§ 12. RAILROAD SURVEYS. 11 

for a back rodman, and the fact that the errors are purely 
local and not cumulative, and may be so limited, with care, that 
they will cause no vital error in the subsequent location survey. 
The transit method is essentially more accurate, but is liable 
to be more laborious and troublesome. If a large tree is en- 
countered, either it must be cut down or a troublesome opera- 
tion of offsetting must be used. If the compass is employed 
under these circumstances, it need only be set up on the far side 
of the tree and the former bearing produced. An error in 
reading a transit azimuth will be carried on throughout the 
survey. An error of only five minutes of arc will cause an off- 
set of nearly eight feet in a mile. Large azimuth errors may, 
however, be avoided by immediately checking each new azimuth 
with a needle reading. It is advisable to obtain true azimuth 
at the beginning of the survey by an observation on the sun* or 
Polaris, and to check the azimuths every few miles by azimuth 
observations. Distances along the backbone line should be 
measured with a chain or steel tape and stakes set ever}^ 100 
feet. When a course ends at a substation, as is usually the case, 
the remaining portion of the 100 feet should be measured along 
the next course. The level party should immediately obtain the 
elevations (to the nearest tenth of a foot) of all stations, and also 
of the loAvest points of all streams crossed and even of dry gullies 
which would require culverts. 

12. Cross-sectioning. It is usually desirable to obtain con- 
tours at five-foot intervals This may readily be done by the 
use of a Locke level (which should be held on top of a simple 
five-foot stick), a tape, and a rod ten feet in length graduated 
to feet and tenths. The method of use may perhaps be best 
explained by an example. Let Fig. 5 represent a section per- 
pendicular to the survey line — such a section as would be made 
by the dotted lines in Fig. 4. C represents the station point. 
Its elevation as determined by the level is, say, 158.3 above 
datum. When the Locke level on its five-foot rod is placed at 
C, the level has an elevation of 163.3. Therefore when a point 
is found (as at a) where the level will read 3.3 on the rod, that 
point has an elevation of 160.0 and its distance from the center 
gives the position of the 160-foot contour. Leaving the long 
rod at that point (a), carry the level to some point (h) such that 
the level will sight at the top of the rod. b is then on the 165- 

* For detailed methods of such determinations, see the authors ' Problems 
in the Use and Adjustment of Engineermg Instruments," Problems 35 and 36. 



12 



RAILROAD CONSTRUCTIOX. 



§12. 



foot contour, and the horizontal distance ah added to the hori- 
zontal distance ac gives the position of that contour from the 
center. The contours on the lower side are found similarl}^ 
The first rod reading will be 8.3, giving the 155-foot contour. 




Fig. 5. 

Plot the results in a note-book which is ruled in quarter-inch 

squares, using a scale of 100 feet per inch in both directions. 

. Plot the work up the page; then when looking ahead along the 

line, the work is properly oriented. When a contour crosses 




Fig. 6. 



the survey line, the place of crossing may be similarly deter- 
mined. If the ground flattens out so that five-foot contours are 
very far apart, the absolute elevations of points at even fifty- 



§ 13. RAILROAD SURVEYS. 13 

foot distances from the center should be determined. The 
method is exceedingly rapid. Whatever error or inaccuracy 
occurs is confined in its effect to the one station where it occurs. 
The work being thus plotted in the field, unusually irregular 
topography may be plotted w^th greater certainty and no great 
error can occur without detection. It w^ould even be possible 
by this method to detect a gross error that might have been 
made by the level party 

13. Stadia method. This method is best adapted to fairly 
open country where a ^'shot" to any desired point may be 
taken without clearing. The backbone survey line is the same 
as in the previous method except that each course is limited to 
the practicable length of a stadia sight. The distance between 
stations should be checked by foresight and backsight — also the 
vertical angle. Azimuths should be checked by the needle. 
Considering the vital importance of leveling on a railroad survey 
it might be considered desirable to run a line of levels over the 
stadia stations in order that the leveling may be as precise as 
possible; but when it is considered that a preliminary survey is 
a somewhat hasty survey of a route that may be abandoned, and 
that the errors of leveling by the stadia method (w'hich are con- 
pensating) may be so minimized that no proposed route would 
be abandoned on account of such small error, and that the effect 
of such an error may be easily neutralized by a slight change in 
the location, it may be seen that excessive care in the leveling 
of the preliminary survey is hardly justifiable. 

Since the students taking this w^ork are assumed to be familiar 
with the methods of stadia topographical surveys, this part of 
the subject will not be further elaborated. 

14. " First " and " Second " preliminary surveys. Some engi- 
neers advocate two preliminary surveys. When this is done, 
the first is a very rapid survey, made perhaps with a compass, 
ard is only a better grade of reconnoissance. Its aim is to 
rapidly develop the facts which will decide for or against any 
proposed route, so that if a route is found to be unfavorable 
another more or less modified route may be adopted without 
having wasted considerable time in the survey of useless details. 
By this time the student should have grasped the fundamental 
idea that both the reconnoissance and preliminary surveys are 
not surveys of lines but of areas; that their aim is to survey 
only those topographical features which w^ould have a deter- 



14 RAILROAD CON6TRUCTIOX. § 15. 

mining influence on any railroad line which might be constructed 
through that particular territory, and that the value of a locating 
engineer is largely measured by his ability to recognize those 
determining influences with the least amount of work from his 
surveying corps. Frequently too little time is spent on the 
comparative study of preliminary lines. A line will be hastily 
decided on after very little study; it will then be surveyed with 
minute detail and estimates carefully worked up, and the claims 
of any other suggested route will then be handicapped, if not 
disregarded, owing to an unwillingness to discredit and throw 
?iway a large amount of detailed surveying. The cost of two or 
three extra preliminary surveys (at critical sections and not over 
the whole line) is utterly insignificant compared wnth the prob- 
able improvement in the ^^ operating value" of a line located 
after such a comparative study of preliminary lines. 

LOCATION SURVEYS. 

15. "Paper location." When the preliminary survey has 
been plotted to a scale of 200 feet per inch and the contours 
drawn in, a study may be made for the location survey. Disre- 
garding for the present the effect on location of transition curves, 
the alignment may be said to consist of straight lines (or 'Han- 
gents") and circular curves. The '^ paper location" therefore 
consists in plotting on the preliminary map a succession of 
straight lines which are tangent to the circular curves connect- 
ing them. The determining points should first be considered. 
Such points are the termini of the road, the lowest practicable 
point over a summit, a river-crossing, etc. So far as is possi- 
ble, having due regard to other considerations, the road should 
be a ''surface" road, i.e., the cut and fill should be made as 
small as possible. The maximum permissible grade must also 
have been determined and duly considered. The method of 
location differs radically according as the lines joining the deter- 
mining points have a very low grade or have a grade that ap- 
proaches the maximum permissible. With very low natural 
grades it is only necessary to strike a proper balance between 
the requirements for easy alignment and the avoidance of exces- 
sive earthwork. When the grade between two determined 
points approaches the maximum, a study of the location may be 
begun by finding a strictly surface line which will connect those 



§ 16. RAILROAD SURVEYS. 15 

points with a line at the given grade. For example, suppose 
the required grade is 1.6% and that the contours are drawn at 
5-foot intervals It will require 312 feet of 1.6% grade to rise 
5 feet. Set a pair of dividers at 312 feet and step off this in- 
terval on successive contours. This line will in general be very 
irregular, but in an easy country it may lie fairly close to the 
proper location line, and even in difficult country such a surface 
line will assist greatly in selecting a suitable location. When the 
larger part of the line will evidently consist of tangents, the tan- 
gents should be first located and should then be connected by 
suitable curves. When the curves predominate, as they gener- 
ally will in mountainous country, and particularly when the line 
is purposely lengthened in order to reduce the grade, the curves 
should be plotted first and the tangents may then be drawn 
connecting them. Considering the ease wnth which such lines 
may be drawn on the preliminary map, it is frequently advisable, 
after making such a paper location, to begin all over, draw a 
new line over some specially difficult section and compare re- 
sults. Profiles of such lines may be readily drawn by noting their 
intersection with each contour crossed. Drawing on each profile 
the required grade line will furnish an approximate idea of the 
comparative amount of earthw^ork required. After deciding on 
the paper location, the length of each tangent, the central angle 
(see § 21), and the radius of each curve should be measured as 
accurately as possible. Since a slight error made in such meas- 
urements, taken from a map with a scale of 200 feet per inch, 
would by accumulation cause serious discrepancies between the 
plotted location and the location as afterward surveyed in the 
field, frequent tie lines and angles should be determined between 
the plotted location line and the preliminary line, and the loca- 
tion should be altered, as may prove necessary, by changing the 
length of a tangent or changing the central angle or radius of a 
curve, so that the agreement of the check-points will be suffi- 
ciently close. The errors of an inaccurate preliminary survey 
may thus be easily neutralized (see § 33). When the pre- 
liminary fine has been properly run, its '* backbone'' line will 
lie very near the location line and will probably cross it at fre- 
quent intervals, thus rendering it easy to obtain short and nu- 
merous tie lines. 

1 6. Surveying methods. A transit should be used for align- 
ment, and only precise work is allowable. The transit stations 



16 



RAILROAD CONSTRUCTION. 



§16. 



should be centered with tacks and should be tied to witness- 
^stakes, which should be located outside of the range of the earth- 
work, so that they will neither be dug up nor covered up. All 
original property lines lying within the limits of the right of w^ay 
should be surveyed w^ith reference to the location line, so that 
the right-of-way agent may have a proper basis for settlement. 
When the property hnes do not extend far outside of the re- 
quired right of way they are frequently surveyed completely. 

The leveler usually reads the target to the nearest thousandth 
of a foot on turning-points and bench-marks, but reads to the 
nearest tenth of a foot for the elevation of the ground at stations. 
Considering that y^^TT of a foot has an angular value of about 



FORM OF NOTES. 



[Left-hand page.] 



Sta. 


Align- 
ment. 


Vernier. 


Tangential 
Deflection. 


Calculated 
Bearing. 


Needle. 


54 












53 

O -^72.2 


P.T. 


9° 11' 


18'' 22' 


N 54° 48' E 


N 62'* 15' E 


52 




7 57 








51 
50 


r- -1 
> c 

^5 


6 15 
4 33 






1 


49 


t) ... 

O 

ccoo 


2 51 








48 


I 1 


1 09 








4-32 
47 


P.O. 


0° 








46 








N 36" 26' E 


N 44° 0' E 



§m. 



RAILROAD SURVEYS. 



17 



one second at a distance of 200 feet, and that one division of a level- 
bubble is usually about 30 seconds, it may be seen that it is a 
useless refinement to read to thousandths unless corresponding 
care is taken in the use of the level. The leveler should also 
locate his bench-marks outside of the range of earthwork. A 
knob of rock protruding from the ground affords an excellent 
mark. A large nail, driven in the roots of a tree, which is not 
to be disturbed, is also a good mark. These marks should be 
clearty described in the note-book. The leveler should obtain 
the elevation of the ground at all station-points; also at all 
sudden breaks in the profile line, determining also the distance 
of these breaks from the previous even station. This will in- 



[ Right -hand page.] 





30=d 


53t60 


( 


) JAS. WILSON 


\^^ I 


^ 


^\ 


52+18 


< 


i^\,^ 




X. 


( 


) 




wm. brown 


. I 


k 


\ i8+75 


48+42 

) 


'^X ^ 


^ JOHN JONES 




) 


> 


16+31 


< 


\ 



18 RAILROAD CONSTRUCTION. § 17. 

elude the position and elevation of all streams, and even dry 
gullies, which are crossed 

Measurements should preferably be made with a steel tape, 
care being taken on steep ground to insure horizontal measure- 
ments. Stakes are set each 100 feet, and also at the beginning 
and end of all curves. Transit-points (sometimes called ^' plugs" 
or ''hubs") should be driven flush with the ground, and a 
''witness-stake," having the '' number " of the station, should 
be set three feet to the right. For example, the witness-stake 
might have on one side "137 + 69.92," and on the other side 
''PC4°R," which would signify that the transit hub is 69.92 
feet beyond station 137, or 13769.92 feet from the beginning of 
the line, and also that it is the *' point of curve" of a '' 4° curve" 
which turns to the right. 

Alignment. The alignment is evidently a part of the loca- 
tion survey, but, on account of the magnitude and importance 
of the subject, it will be treated in a separate chapter. 

17. Form of Notes. Although the Form of Notes cannot be 
thoroughly understood until after curves are studied, it is here 
introduced as being the most convenient place. The right-hand 
page should have a sketch showing all roads, streams, and 
property lines crossed with the bearings of those lines. This 
should be drawn to a scale of 100 feet per inch — the quarter- 
inch squares which are usuall}^ ruled in note-books giving con- 
venient 25-foot spaces This sketch will always be more or less 
distorted on curves, since the center line is always shown as 
straight regardless of curves. The station points ('' Sta." in 
first column, left-hand page) should be placed opposite to their 
sketched positions, which means that even stations will be 
recorded on every fourth line. This allows three intermediate 
lines for substations, which is ordinarily more than sufficient. 
The notes should read up the page, so that the sketch will be 
properly oriented when looking ahead along the line The 
other columns on the left-hand page will be self-explanatory 
when the subject of curves is understood. If the ''calculated 
bearings" are based on azimuthal observations, their agreement 
(or constant difference) with the needle readings will form a 
valuable check on the curve calculations and the instrumental 
work. 



CHAPTER II. 



ALIGNMENT. 



In this chapter the alignment of the center line only of a 
pair of rails is considered. When a railroad is crossing a sum- 
mit in the grade hne, although the horizontal projection of the 
alignment may be straight, the vertical projection Avill consist of 
two sloping lines joined bj^ a curve. When a curve is on a 
grade, the center line is realh' a spiral, a curve of double curva- 
ture, although its horizontal projection is a circle. The center 
line therefore consists of straight lines and curves of single 
and double curvature. The simplest method of treating them 
is to consider their horizontal and vertical projections separately. 
In treating simple, compound, and transition curves, only the 
horizontal projections of those curves will be considered. 



SIMPLE CURVES. 

1 8. Designation of curves. A curve ma}^ be designated either 
by its radius or by the angle subtended by a chord of unit 
length. Such an angle is known 
as the ^' degree of curve " and is 
indicated by D. Since the curves 
that are practically used have very 
long radii, it is generally impracti- 
cable to make any use of the actual 
center, and the curve is located 
without reference to it. If AB in 
Fig. 4 represents a unit chord (C) 
of a curve of radius R, then by the 
above definition the angle AOB 
equals D, Then 




AOs>\n\D = \AB = \C. 



R = 



sin JD' 



(1) 



19 



20 



RAILROAD CONSTRUCTION. 



§19. 



or, by inversion, 



sin iD = 



2/r 



(2) 



The unit chord is variously taken throughout the world as 
100 feet, 66 feet, and 20 meters. In the United States 100 
feet is invariably used as the unit chord length, and throughout 
this work it will be so considered. Table I has been computed 
on this basis. It gives the radius, Avith its logarithm, of all 
curves from a 0° 01' curve up to a 10° curve, varying by single 
minutes. The sharper curves, which are seldom used, are given 
Avith larger intervals. 

An approximate value of R may be readily found from the 
following simple rule, which should be memorized: 



7?- 



5730 



D 



Although such values are not mathematically correct, since R 
does not strictly vary in\'ersely as D, yet the resulting value is 
within a tenth of one per cent for all commonly used values 
of R, and is sufficiently close for many purposes, as will be 
shown later. 

19. Length of a subchord. Since it is impracticable to 
measure along a curved arc, curves are always measured by 

laying off 100-foot chord lengths. 
This means that the actual arc is 
alwavs a little longer than the 
chord. It also means that a sub- 
chord (a chord shorter than the unit 
length) will be a little longer than 
the ratio of the angles subtended 
Avould call for. The truth of this 
may be seen w^ithout calculation 
by noting that two equal sub- 
chords, each subtending the angle 
Fig. 8. JD, will evidently be slightly longer 

than 50 feet each. If c be the length of a subchord subtend- 
ing the angle d, then, as in Eq. 2, 




sin ^d -- 



2R' 



§ 20. ALIGNMENT. 21 

or, by inversion, 

C'=2Rsmid (3) 

The nominal length of a subchord= 100^: For example, 

a nominal subchord of 40 feet will subtend an angle of -j^"^ of 
Z)°; its true length will be slightly more than 40 feet, and may 
be computed by Eq. 3. The difference between the nominal 
and true lengths is maximum when the subchord is about 57 
feet long, but with the low degrees of curvature ordinarily used 
the difference may be neglected. With a 10° curve and a 
nominal chord length of 60 feet, the true length is 60.049 feet. 
Very sharp curves should be laid off with 50-foot or even 25- 
foot chords (nominal length). In such cases especially the true 
lengths of these subchords should be computed and used instead 
of the nominal lengths. 

20. Length of a curve. The length of a curve is always 
indicated by the quotient of 100J-^Z). If the quotient of 
J^D is a whole number, the length as thus indicated is the 
true length — measured in 100-foot chord lengths. If it is an 
odd number or if the curve begins and ends with a subchord 
(even though J-=-Z) is a whole number), theoretical accuracy 
requires that the true subchord lengths shall be used, although 
the difference may prove insignificant. The length of the arc 
(or the mean length of the two rails) is therefore always in 
excess of the length as given above. Ordinarily the amount 
of this excess is of no practical importance. It simply adds an 
insignificant amount to the length of rail required. 

Example. Required the nominal and true lengths of a 
3° 45' curve having a central angle of 17° 25'. First reduce 
the degrees and minutes to decimals of a degree. (100 X 1 7° 25') 
--3° 45' = 1741.667 -^3.75 =464.444. The curve has four 100- 
foot chords and a nominal chord of 64.444 The true chord 
should be 64.451, The actual arc is 

17°.4167 X j|^o X /^ = 464.527 

The excess is therefore 464.527 -464.451 =0.076 foot. 

21. Elements of a curve. Considering the line as running 
from A toward B, the beginning of the curve, at A, is called 
the point of curve {PC). The other end of the curve, at B, is 



22 



RAILROAD CONSTRUCTION, 



§22. 



called the point of tangency (PT). The intersection of the 

tangents is called the vertex (V). 
The angle made by the tangents 
at F, which equals the angle 
made by the radii to the extrem- 
ities of the curve, is called the 
central angle (J). AV and BV, 
the two equal tangents from the 
vertex to the PC and PT, are 
called the tangent distances (T). 
The chord AB is called the long 
^ chord {LC). The intercept HG 
from the middle of the long 
chord to the middle of the arc 
is called the middle ordinate (M). 
^'°- ^' That part of the secant GV from 

the middle of the arc to the vertex is called the external distance 
(E). From the figure it is very easy to derive the following fre- 
quently used relations: 

T = RtSiniJ (4) 

LC = 2R sin i J (5) 

M = RveTsiJ (6) 

E = RexseQ y (7) 

22. Relation between T, E, and J. Join A and G in Fig. 9. 
The angle VAG = \J, since it is measured by one half of the 
arc AG between the secant and tangent AGO=90° — JJ. 




sin AGV 



AV :VG : : sin AGV : sin VAG; 
= sin AGO = cos JJ; 

T : E : : cos IJ : sin JJ; 
T = ^cotJJ 



(8) 



The same relation may be obtained by dividing Eq. 4 by Eq. 
7, since tan a -^exsec a = cot Ja. 

23. Elements of a 1° curve. From Eqs. 1 to 8 it is seen that 
the elements of a curve vary directly as R. It is also seen to 
be very nearly true that R varies inversely as D. If the ele- 
ments of a 1° curve for various central angles are calculated and 
tabulated, the elements of a curve of D° curvature may be 
approximately found by dividing by D the corresponding ele- 
ments of a 1° curve having the same central angle. For small 



§24. 



ALIGNMENT. 



23 



central angles and low degrees of curvature the errors involved 
by the approximation are insignificant, and even for larger 
angles the errors are so small that for many purposes they may be 
disregarded 

In Table II is given the value of the tangent distances, 
external distances, and long chords for a 1° curve for various 
central angles The student should familiarize himself with the 
degree of approximation involved by solving a large number of 
cases under various conditions by the exact and by the approxi- 
mate methods, in order that he may know when the approxi- 
mate method is sufficiently exact for the intended purpose. 
The approximate method also gives a ready check on the 
exact method. 

24. Exercises, (a) What is the tangent distance of a 4° 20' 
curve having a central angle of 18° 24'? 

(b) Given a 3° 30' curve and a central angle of 16° 20', how 
far will the curve pass from the vertex? [Use Eq. 7.] 

(c) An 18° curve is to be laid off using 25-foot (nominal) 
chord lengths. What is the true length of the subchords? 

(d) Given two tangents making a central angle of 15° 24'. 
It is desired to connect these tangents by a curve which shall 
pass 16.2 feet from their intersection. How far down the 
tangent will the curve begin and what will be its radius? (Use 
Eq. 8 and then use Eq 4 inverted.) 

25. Curve location by deflections. The angle between a 
secant and a tangent (or between two secants intersecting on an 
arc) is measured by one half of the intercepted arc. Beginning 
at the PC (A in Fig. 10), if the 
first chord is to be a full chord 
we may deflect an angle VAa 
(=^D), and the point a, which is 
100 feet from .4 , is a point on the 
curve. For the next .station, h, 
deflect an additional angle bAa 
(=^D) and, with one end of the 
tape at a, swing the other end 
until the 100-foot point is on the 
line Ab. The point b is then on 
the curve. If the final chord cB 
is a subchord, its additional deflec- Fig. 10. 

tion (^d) is something less than JD. The last deflection (BA V) is 




24 RAILROAD CONSTRUCTION. § 26. 

of course JJ. It is particularly important, when a curve begins 
or ends with a subchord and the deflections are odd quantities, 
that the last additional deflection should be carefully com- 
puted and added to the previous deflection, to check the mathe- 
matical work by the agreement of this last computed deflec- 
tion with JJ. 

Example. Given a 3° 24' curve having a central angle of 
18° 22' and beginning at sta. 47 + 32, to compute the deflec- 
tions. The nominal length of curve is 18° 22' ^3° 24' = 18.367^ 
3.40 = 5.402 stations or 540.2 feet. The curve therefore ends 
at sta. 52 + 72.2. The deflection for sta. 48 is iVoXi(3°24') 
=0.68Xl°.7 = l°. 156 = 1° 09' nearly. For each additional 100 
feet it is 1° 42' additional. The final additional deflection for 
the final subchord of 72.2 feet is 

^ X i(3° 24') = 1°.2274 = 1° 14' nearly. 

The deflections are 

P. C . . . Sta. 47 + 32 ' 0° 

48 0° +1°09' = 1°09' 

49 1°09' + 1°42'=2°51' 

50 2° 51' + 1° 42' =4° 33' 

51 4°33' + l°42'=6°15' 

52 6° 15' + 1° 42' = 7° 57' 

P. T 52 + 72.2 7°57' + l° 14'=9°11' 

As a check 9° ll' = i(18° 22') =iz/. (See the Form of Notes 
in § 17.) 

26. Instrumental work. It is generally impracticable to 
locate more than 500 to 600 feet of a curve from one station. 
Obstructions will sometimes require that the transit be moved up 
every 200 or 300 feet. There are two methods of setting off 
the angles when the transit has been moved up from the PC. 

(a) The transit may be sighted at the previous transit station 
with a reading on the plates equal to the deflection angle from 
that station to the station occupied, but with the angle set off on 
the other side of 0°, so that when the telescope is turned to 0° it 
will sight along the tangent at the station occupied. Plunging 
the telescope, the forward stations may be set off by deflecting 
the proper deflections from the tangent at the station occupied 



§ 26. 



ALIGNMENT. 



25 



This is a very common method and, when the degree of curva- 
ture is an even number of degrees and when the transit is onl}- 
set at even stations, there is but little objection to it. But the 
degree of curvature is sometimes an odd quantity, and the exi- 
gencies of difficult location frequently require that substations 
be occupied as transit stations. Method (a) will then require 
the recalculation of all deflections for each new station occupied. 
The mathematical work is largely increased and the probability 
of error is very greatly increased and not so easily detected. 
Method (b) is just as simple as method (a) even for the most 
simple cases, and for the more difficult cases just referred to the 
superiority is very great. 

(b) Calculate the deflection for each station and substation 
throughout the curve as though the whole curve were to be lo- 
cated from the PC. The computations 
may thus be completed and checked (as 
above) before beginning the instrumental 
work. If it unexpectedly becomes neces- 
sary to introduce a substation at any 
point, its deflection from the PC may be 
readily interpolated. The stations actually 
set from the PC are located as usual. 
Rule. When the transit is set on any 
forward station, backsight to any previous 
station with the plates set at the deflection 
angle for the station sighted at. Plunge 
the telescope and sight at any forward 
station with the deflection angle originally 
computed for that station. When the 
plates read the deflection angle for the 
station occupied, the telescope is sighting 
along the tangent at that station — which 
is the method of getting the forward tan- 
gent when occupying the PT. Even though 
the station occupied is an unexpected sub- 
station, when the instrument is properly 
oriented at that station, the angle reading 
for any station, forward or back, is that originally computed 
for it from the PC. In difficult work, where there are ob- 
structions, a valuable check on the accuracy may be found by 
sighting backward at any visible station and noting whether 



2-6 



RAILROAD CONSTRUCTION. 



26. 



its deflection agrees with that originally computed. As a 
numerical illustration, assume a 4° curve, with 28° curvature, 
with stations 0, 2, 4, and 7 occupied. After setting stations 
1 and 2, set up the transit at sta. 2 and backsight to sta. 
with the deflection for sta. 0, which is 0°. The reading on sta. 
1 is 2°; when the reading is 4° the telescope is tangent to 
the curve, and when sighting at 3 and 4 the deflections will be 
6° and 8°. Occupy 4; sight to 2 with a reading of 4°. When 
the reading is 8° the telescope is tangent to the curve and, by 
plunging the telescope, 5, 6, and 7 may be located with the 
originally computed deflections of 10°, 12°, and 14°. When oc- 
cupying 7 a backsight may be taken to any visible station with 
the plates reading the deflection for that station; then when 





Fig. 12. 



Fig. 13. 



the plates read 14° the telescope will point along the forward 
tangent. 

The location of curves by deflection angles is the normal 
method. A few other methods, to be described, should be con- 
sidered as exceptional. 



§27. 



ALIGNMENT. 



27 



27. Curve location by two transits. A curve might be located 
more or less on a swamp where accurate chaining would be 
exceedingly difficult if not impossible. The long chord AB 
(Fig. 12) may be determined by triangulation or otherwise, 
and the elements of the curve computed, including (possibly) 
subchords at each end. The deflection from A and B to each 
point may be computed. A rodman may then be sent (by 
whatever means) to locate long stakes at points determined 
by the simultaneous sightings of the two transits. 

28. Curve location by tangential offsets. When a curve is 
very flat and no transit is at hand the following method mav be 
used (see Fig. 13) : Produce the back tangent as far forward as 
necessary. Compute the ordinates Oa\ Ob\ Oc^, etc., and the 
abscissae a^a, h'b, c'c, etc. If Oa is a full station (100 feet), then 



Oa'^Oa' =100 cos iZ), s\so = RsmD; 

Oy=Oa'-\-a'y = 100 cos iD + 100 cos ID, 

also = R sin 2D ; 
Oc'=Oa' + a'b' + h'c' = 100(cos JD + cos |Z) + cos fD), 

also = i? sin 3D; 



etc. 



a^a = 100 sin JD, also = R vers D ; 

b'b =a'a-\-b"b =100 sin W -f 100 sin |i), 

also = R vers 2D ; 
c'c = Vb + c''c = 100(sin \D -f sin fZ) -f- sin |D) , 

also = R vers 3Z) ; 



(9) 



(10) 



etc. 

The functions JD, |D, etc., may be more conveniently used 
without logarithms, by adding the several natural trigonometrical 
functions and pointing off two decimal places. It may also be 
noted that Ob' (for example) is one half of the long chord for 
four stations; also that b'b is the middle ordinate for four 
stations. If the engineer is provided with tables giving the long 
chords and middle ordinates for various degrees of curvature, 
these quantities may be taken (perhaps by interpolation) from 
such tables. 

If the curve begins or ends at a substation, the angles and 
terms will be correspondingly altered. The modifications may 



28 RAILROAD CONSTRUCTION. § 29. 

be readily deduced on the same principles as above, and should 
be worked out as an exercise by the student. 

In Table II are giv^en the long chords for a 1° curve for various 
values of J. Dividing the value as given b}^ the degree of the 
curve, we have an approximate value which is amply close for 
low degrees of curvature, especially for laying out curves with- 
out a transit. For example, given a 4° 30' curve, required the 
ordinate Oc\ This is evidently one half of a chord of six stations, 
with J =27°. Dividing 2675.1 (which is the long chord of a 
1° curve with J =27°) by 4.5 we have 594.47; one half of this is 
the required ordinate, Oc' =297.23. The exact value is 297.31, 
an excess of .08, or less than .03 of 1%. The true values 
are always slightly in excess of the value as computed from 
Table II. 

Exercise. A 3° 40' curve begins at sta. 18 + 70 and runs to 
sta. 23 + 60. Required the tangential offsets and their corre- 
sponding ordinates. The first ordinate = 30 cos i( tVu X 3° 40') = 
30 X. 99995 =29.9985; the offset = 30 sin 0° 33' = 30 X. 0096 = 
0.288. For the second full station (sta. 20) the ordinate = 
i long chord for J =2(1° 06' + 3° 40') with Z)=3°40'. Divid- 
ing 476.12, from Table II, by 3 J, we have 129.85. Otherwise, 
by Eq. 9, the ordinate = 30 X cos 0° 33' + 100 cos (1° 06' + l° 50') 
= 30.00 + 99.87 = 129.87. The offset for sta. 20 = 30 sin 0° 33' + 
100 sin (1° 06' + 1° 50') = 0.288 + 5.12 =5.41. Workout 
similarly the ordinates and offsets for sta. 21, 22, 23, and 
23 + 60. 

29. Curve location by middle ordinates. Take first the sim- 
pler case when the curve begins at an even station. If we con- 
sider (in Fig. 14) the curve produced back to z, the chord za = 
2 X 100 cos JD, A'a = 100 cos JD, and A'A=am = zn = 100 sin JD. 
Set off ^A' perpendicular to the tangent and A 'a parallel to 
the tangent. AA' =aa' =66' =cc', etc. = 100 sin JZ). Set off 
aa' perpendicular to a' A. Produce Aa' until a^b^A'a^ thus 
determining h. Succeeding points of the curve may thus be 
determined indefinitely. 

Suppose the curve begins with a subchord. As before 
ra = Am' =c' cos ^d\ and rA =a7n' =c^ sin \d\ Also s2 = An' = 
c"cosK', and sA =2^'=c" sin K', in which (d' + c?")=D. 
The points z and a being determined on the ground, aa' may 
be computed and set off as before and the curve continued in 



§ 30. 



ALIGNMENT. 



29 



full stations. A subchord at the end of the curve may be located 
by a similar process. 

30. Curve location by offsets from the long chord. (Fig. 16.) 
Consider at once the general case in which the curve commences 
with a subchord (curvature, d')j continues with one or more full 




Fig. 14. 



Fig. 15. 



Fig. 16. 



chords (curvature of each, D), and ends with a subchord with 
curvature d". The numerical work consists in computing first 
AB, then the various abscissae and ordinates. AB=2R sin J J. 



Ah' = Aa' + a'h' =</ cos K^ -<^') + 100 cos K^ -2d'-Z)); 

Ac =Aa'-\-a'V + h'c^ = </ cos hi J- d') + \00 cos ^( J- 2d'- D) 

-\-\0Q cos ^{J -2d" -D)\ 
also 

^AB-B(^ ^2Rs\nhA-c" cos^id^d"), 

a'a^a'a —c'sin^iA—d'); 

h'h=^a'a + mh = c' sin i(J -d')+*lOO sin \{A-2d'-D)\ 

c'c =b'b-nh = c' sin i(J -d') + 100 sin hiJ -2d' ~D) 

-100 sin i( J- 2d'' -D); 
also ^c" sin UJ-d"). 



(11) 



V (12) 



The above formulae are considerably simpHfied when the 



30 



RAILROAD CONSTRUCTION. 



§31. 



curve begins and ends at even stations. When the curve is 
very long a regular law becomes very apparent in the formation 
of all terms between the first and last. There are too few terms 
in the above equations to show the law. 

31. Use and value of the above methods. The chief value 
of the above methods lies in the possibility of doing the work 
without a transit. The same principles are sometimes em- 
ployed, even when a transit is used, when obstacles prevent the 
use of the normal method (see § 32, c). If the terminal tan- 
gents have already been accurately determined, these methods 
are useful to locate points of the curve when rigid accuracy is 
not essential. Track foremen frequently use such methods to 
lay out unimportant sidings, especially when the engineer and 
his transit are not at hand. Location by tangential offsets (or 
by offsets from the long chord) is to be preferred when the 
curve is flat (i.e., has a small central angle J) and there is no 
obstruction along the tangent, or long chord. Location by 
middle ordinates may be employed regardless of the length of 
the curve, and in cases when both the tangents and the long 
chord are obstructed. The above methods are but samples 
of a large number of similar methods which have been devised. 
The choice of the particular method to be adopted must be 
determined by the local conditions. 

32. Obstacles to location. In this section will be given only 
a few of the principles involved in this 
class of problems, with illustrations. The 
engineer must decide, in each case, which 
is the best method to use. It is frequently 
advisable to devise a special solution for 
some particular case. 

a. When the vertex is inaccessible. As 
shown in § 26, it is not absolutely essential 
that the vertex of a curve should be 
located on the ground. But it is very evi- 
dent that the angle between the terminal 
tangents is determined with far less prob- 
able error if it* is measured by a single 
measurement at the vertex rather than as 
the result of numerous angle measurements 
FiQ. 17. along the curve, involving several posi- 

tions of the transit and comparatively short sights Some- 




^ 



§ 32. ALIGNMENT. 31 

times the location of the tangents is abeady determined on 
the ground (as by hn and am, Fig. 17), and it is required to 
join the tangents by a curve of given radius. Method. Measure 
ah and the angles Vba and baV. A is the sum of these angles. 
The distances hV and aV are computable from the above data. 
Given A and R, the tangent distances are computable, and then 
Bh and a A are found by subtracting hV and aV from the tan- 
gent distances. The curve may then be run from A, and the 
T/ork may be checked by noting whether the curve as run ends 
at B — previously located from h. 

Example. Assume ah =54:6 82; angle a = 15° 18'; angle 
b == 18° 22' ; D= 3° 40' ; required a A and bB. 
J = 15°18' + 18°22'=33°40' 

Eq. (4) R (3° 40') 3.19392 

tan y =tan 16° 50' 9.48080 

r=472.85 2.67475 

v= xSin_18^' ab 2.73784 

®^ sin 33° 40' log sin 18° 22' 9.49844 

co-log sin 33° 40' 0.25621 

aF=310.81 2.49250 

A F =472. 85 

aA =162.04 



^ sin 33° 40' log sin 15° 18' 9.42139 

co-log sin 33° 40' 0.25621 

6F=260.29 2.41545 

57=472.85 

65=212.56 



b. When the point of curve (or point of tangency) is inacces- 
sible. M some distance (As, Fig. 18) an unobstructed line pn 
may be run parallel with AV. nv=py=As=R vers a. 

'. vers a=As-7-R. 

ns=ps^R sin a^ 



32 



RAILROAD COXSTRUCTIOX. 



§33. 



At 2/, which is at a distance ps back from the computed posi- 
tion of Ay make an offset sA 
to p. Run pn parallel to the 
tangent. A tangent to the 
curve at n makes an angle of a 
with np. From n the curve is 
run in as usual 

If the point of tangency is 
obstructed, a similar process, 
somewhat reversed, may be 
used. /? is that portion of J still 
to be laid off when m is reached. 
tm=tl=R sin ^. mz—tB=lx=R 
vers 3. 

c. When the central part of 

the curve is obstructed, a is the 

central angle between two points 

of the curve between which 

a may equal any angle, but it is prefer- 




FiG. 18. 



a chord ma}' be run. 
able that a should be a multiple 
of Df the degree of curve, and that 
the points m and n should be on 
even stations. mn=2Rs'mha. A 
point s may be located by an offset 
ks from the chord mn by a similar 
method to that outlined in § 30. 

The device of introducing the 
dotted curve mn having the same 
radius of curvature as the other, 
although neither necessary nor 
advisable in the case shown in 
Fig. 19, is sometimes the best 
method of surveying around an 
obstacle> The offset from any point on the dotted curve to 
the corresponding point on the true curve is twice the " ordinate 
to the long chord," as computed in § 30. 

33. Modifications of location. The following methods may 
be used in allowing for the discrepancies between the ^' paper 
location" based on a more or less rough preliminary survey and 
the more accurate instrumental location. (See § 15.) They are 




Fig. 19. 



§ 33. 



ALIGNMENT. 



33 



also frequently used in locating new parallel tracks and modify- 
ing old tracks. 

a. To move the forward tangent parallel to itself a distance », 
the point of curve (a) remaining fixed. (Fig. 20.) 



VV'=^ 






~smhVV'~smJ 

AV'=AV + VV\ 
The triangle BniB^ is isosceles and Bm=B^m, 



vers B^mB vers A' 



(13) 



/. R'=R + 



vers J* 



(14) 



The solution is very similar in case the tangent is moved in- 
ward to yj5'^ Note that this method necessarily changes the 




Fig. 20. 




Fig. 21. 



radius. If the radius is not to be changed, the point of curve 
must be altered as follows: 

b. To move the forward tangent parallel to itself a distance x, 
the radius being unchanged. (Fig. 21.) In this case the whole 



34 



RAILROAD CONSTRUCTION. 



§34. 



curve is moved bodily a distance 00^ =AA' = VV'=BB^, and 
moved parallel to the first tangent A V 
B'n X 



BB'=^- 



r^AA'. 



(15) 



'sin nBB' sin A 

c. To change the direction of the forward tangent at the point 
of tangency. (Fig. 22.) This problem involves a change (a) in 

the central angle and also requires a 
new radius. An error in the deter- 
mination of the central angle fur- 
nishes an occasion for its use. 
R, A J a J AVf and BV are known. 
J' = J-a. 




Bs=R vers J, 
.-. R'=R- 



Bs=R' vers Jr 
vers J 



(16) 



Fig. 22. 



vers {J — a)' 
As-=R sin J. A's==R' sin J\ 
.'. AA'^A's-As^R' sin J' -R sin J. . . (17) 

The above solutions are given to illustrate a large class of 
problems which are constantly arising. All of the ordinary 
problems can be solved by the application of elementary geome- 
try and trigonometry. 

34. Limitations in location. It may be required to run a 
curve that shall join two given tangents and also pass through a 
given point The point (P, Fig. 
23) is assumed to be deter- 
mined by its distance (VP) 
from the vertex and by the 
angle A7P=/?. 

It is required to determine 
the radius (R) and the tangent 
distance (AV). A is known. 
PF(y = J( 180° -i)-/? 
= 90°-(Ji+.5). 
PP' = 2FP sin P7G 
=27Pcos(ii+/?). 




SP=^VP 



sin/? 
sin Jj' 



Fig. 23. 



§ 35. ALIGNMENT. 35 

AS = VSPXSP' = VSP(SP + PP^ 

lvp4^[vP-^^2VPcosiU+,3)'\ 
\ sin JJ [^ sin iJ J 



^n^ 2 sin /? cos (jJ +^) 
\sin2 J J sin J J 



Sin J J 

= 4^[sin(ii4-^)4-Vsin2^ + 2sin/?siniJcos(iJ+/?)]. (18) 

R=AV cot i J. 

In the special case in which P is on the median line OV, 
fi^gO^'-iJ, and (JJ+/?)=90°. Eq. 18 then reduces to 

AV^-X^n +COS JJ) =FP cot iJ, 

as might have been immediately derived from Eq. 8. 

In case the point P is given by the offset PK and by the 
distance VK, the triangle PKV may be readily solved, giving the 
distance VP and the angle /?, and the remainder of the solution 
will be as above. 

35. Determination of the curvature of existing track, (a) Using 
a transit Set up the transit at any point in the center of the 
track. Measure in each direction 100 feet to points also in the 
center of the track. Sight on one point with the plates at 0°. 
Plunge the telescope and sight at the other point. The angle 
between the chords equals the degree of curvature. 

(b) Using a tape and string. Stretch a string (say 50 feet 
long) between two points on the inside of the head of the outer 
rail. Measure the ordinate (x) between the middle of the string 
and the head of the rail. Then 

_, chord^ , , ^ 

-K=—g^( very nearly) (19) 

For, in Fig. 24, since the triangles AOE and ADC are similar, 




36 RAILROAD CONSTRUCTION. § 3G. 

AO :AE ::AD -.DC or R = hAD'-^x. When, as is usual, 
the arc is very short compared with the 
radius, AD=^AB, very nearly. Making 
this substitution we have Eq. 19. With a 
chord of 50 feet and a 10° curve, the result- 
ing difference in x is .0025 of an inch — far 
within the possible accuracy of such a 
method. The above method gives the 

radius of the inner head of the outer rail. 
Fig 24 

It should be diminished by ^g for the radius 

of the center of the track. ^ With easy curvature, however, this 
will not affect the result by more than one or two tenths of one 
per cent. 

The inversion of this formula gives the required middle or- 
dinate for a rail on a given curve. For example, the middle 
ordinate of a 30-foot rail, bent for a 6° curve, is 

a: = 900 -- (8 X 955) =.118 foot = 1.4 inches. 

Another much used rule is to require the foreman to have a 
string, knotted at the center, of such length that the middle 
ordinate, measured in inches, equals the degree of curve. To 
find that length, substitute (in Eq. 19) 5730^1) for R and 
Z)-hl2 for X. Solving for chord, we obtain chord = Ql.8 feet. 
The rule is not theoretically exact, but, considering the uncertain 
stretching of the string, the error is insignificant. In fact, the 
distance usually given is 62 feet, which is close enough for all 
purposes for which such a method should be used. 

36. Problems. A systematic method of setting down the 
solution of a problem simplifies the work. Logarithms should 
ahvays be used, and all the work should be so set down that a 
revision of the work to find a supposed error may be readily 
done. The value of such systematic work will become more 
apparent as the problems become more complicated. The two 
solutions given below will illustrate such work. 

a. Given a 3° curve beginning at Sta. 27 + 60 and running 
to Sta. 32 + 45. Compute the ordinates and offsets used in 
locating the curve by tangential offsets. 

b. With the same data as above, compute the distances to 
locate the curve by offsets from the long chord. 

c. Assume that in Fig, 17 ab is measured as 217.6 feet, the 



§ 36. ALIGNMENT. 37 

angle abV = 17° 42', and the angle 6aF = 21° 14'. Join the 
tangents by a 4° 30' curve. Determine hB and a A. 

d. Assume that in a case similar to Fig. 18 it was noted 
that a distance (As) equal to 12 feet would clear the building. 
Assume that J =38° 20' and that Z)=4°40'. Required the 
value of a and the position of n. Solution: 

vers a = As-i-R ^s = 12 log = l. 07918 

R (for 4° 40' curve) log = 3.08928 

a=8°01' log vers a = 7.98991 

ns=i^sina log sin a = 9. 14445 

log 7^ = 3. 08923 
ng = 171.27 log = 2 . 23369 

€. Assume that the forward tangent of a 3° 20' curve having 
a central angle of 16° 50' must be moved 3.62 feet inward, with- 
out altering the P.C. Required the change in radius. 

/, Given two tangents making an angle of 36° 18'. It is 
required to pass a curve through a point 93.2 feet from the 
\ ertex, the line from the vertex to the point making an angle 
of 42^^ 21' with the tangent. Required the radius and tangent 
distance. Solution: Applying Eq. 18, we have 

2 log= 0.30103 

./9 = 42°21' log sin = 9.82844 

ii = 18°09' log sin = 9.49346 

(iJ+}) =60° 30' log cos = 9.69234 

.20667 9.31527 

log sin^ /9 = 9 . 65688 .45382 

_2|9. 81987 .66049 

9.90993....' .81271 
nat. sin 60° 30' .8708 

1.6830 log= 0.22610 

yP = 93.2 log = 1.96941 

2.19551 
log sin iJ = 9.49346 

Tang, dist. .4 7 = 503.36 Jog= 2.70205 

log cot i J = 10.48437 

i^ = 1536.1 log= 3.18642 

i)=3°44' 



38 



RAILROAD CONSTRUCTION. 



37. 



COMPOUND CURVES. 

37. Nature and use. Compound curves are formed by a 
succession of two or more simple curves of different curvature. 
The curves must have a common tangent at the point of com- 
pound curvature (P.C.C.). In mountainous regions there is 
frequently a necessity for compound curves having several 
changes of curvature. Such curves may be located separately 
as a succession of simple curves, but a combination of two 
simple curves has special properties which are worth investigat- 
ing and utilizing. In the following demonstrations R2 always 
represents the longer radius and 7?i the shorter j no matter which 
succeeds the other. T^ is the tangent adjacent to the curve of 
shorter radius (Ri), and is invariably the shorter tangent. J^ is 
the central angle of the curve of radius Ri, but it may be greater 
or less than J2 

38. Mutual relations of the parts of a compound curve having 
two branches. In Fig. 25, AC and CB are the two branches of 




Fig. 25. 



the compound curve having radii of Ri and R2 and central angles 
of Ji and Jj- Produce the arc AC to n so that AOin = J. The 
chord Cn produced must intersect B. The line ns, parallel to 
CO2, will intersect BO2 so that Bs=sn = 020i=R2—Ri. Draw 
Am perpendicular to O^n. It will be parallel to hk. 



§ 38. ALIGNMENT. 39 

Br = sn vers Bsn ={R2 — ^i) vers Jj ; 
mn = AOi veis AO^n =Ri vers J; 
Ak=AV sin A Vk = Tj sin J ; 
^ A; = /im = mn -{-nh= mn + 5r. 
.*. Ti sin J =i^i vers J + {R2-R1) vers Jj- • . (20) 
Similarly it may be shown that 

T2 sin J =i?2 vers d-^R^-R^) vers J^. . . (21) 

The mutual relations of the elements of compound curves 
may be solved by these two equations. For example, assume 
the tangents as fixed (J therefore known) and that a curve of 
given radius R^ shall start from a given point at a distance Tj 
from the vertex, and that the curve shall continue through a 
given angle A^. Required the other parts of the curve. From 
Eq. 20 we have 

7^1 sin J — 7?i vers d 
vers J2 

., R^^R^^h^'-J^R^J^ (22) 

vers(J — Jj) ^ 

T2 may then be obtained from Eq. 21. 

As another problem, given the location of the two tangents, 
with the two tangent distances (thereby locating the PC and 
PT)f and the central angle of each curve; required the two 
radii. Solving Eq. 20 for R^, we have 

-. T^i sin J— 7^2 vers ^2 
l^ — — ^ 

vers J — vers J 2 
Similarly from Eq. 21 we may derive 

^ T2 sin d — i?2(vers J — vers J,) 

ji = , 

vers Jj 

Equating these, reducing, and solving for R2, we have 

7^1 sin d vers Jj — 7^2 sin d (vers d — vers d^ 

^~~ vers ^2 vers Jj — (vers J —vers Ji)(vers d —vers dj)' 

Although the various elements may be chosen as above with 

considerable freedom, there are limitations. For example, in 

Eq. 22, since R2 is always greater than R^, the term to be 

added to R^ must be essentially positive — i.e., Tj sin d must be 

vers d J 

greater than R. vers d. This means that T^>R.— — 7-, or that 

sin d 



40 



RAILROAD CONSTRUCTION. 



§ 39. 



T'i>/?i tan Ji, or that T^ is greater than the corresponding 
tangent on a simple curve. Similarly it may be shown that 7\ 
is less than R2 tan JJ or less than the corresponding tangent 
on a simple curve. Nevertheless T2 is always greater than T^. 
In the limiting case when 7?2=^i» ^2 = ^1^ ^^d ^2 = ^1- 

39. Modifications of location. Some of these modifications 
may be solved by the methods used for simple curves. For 
example* 

a. It is desired to move the tangent VB, Fig. 20, parallel to 
itself to V^B\ Run a new curve from the P.C.C. which shall 
reach the new tangent at B^, where the chord of the old curve 





Fig. 26. 



Fig. 27. 



intersects the new tangent. The solution is almost identical 
with that in § 38, n. 

b. Assume that it is desired to change the forward tangent 
(as above) but to retain the same radius. In Fig. 27 

{R2 — R1) cos J 2 =02n; 

(R2-R1) cos J/ =02V. 



X=02' 



-O2V =(i?2-^l)(C0S J2-COS J/). 
X 



COS Jo' = COS Jo 



(24) 



J2— '^ua^2 — ^ ^ 

II2 — /ti 

The P.C.C, is moved backward along the sharper curve an 
angular distance of J2'~^2 = ^i~^/- 

In case the tangent is moved inward rather than outward, 
the solution will apply by transposing J 2 ^^^ ^i - Then we 
shall have 

X 



cos Jo' = cos ^9 + 



7?2 "~-^l 



(26) 



§30. 



ALIGNMENT. 



41 



The P.C.C. is then moved forward. 

c. Assume the same case as (b) except that the larger radius 
comes first and that the tangent adjacent to the smaller radius 
is moved. In Fig. 28 

(R2—R1) cos J^ =0{n; 
{R2-R1) cosii'=OiV. 



x=0,'n'-0,n 

cos i/ = COS ii + 



R2 — RI 



-COS Jj). 



(26) 




The P.C.C. is moved forward 
along the easier curve an angular 
distance of J/ — ii = J2 — ^2'- 

In case the tangent is moved inward j transpose as before and 
we have 

X 



Fig. 28. 



cos J/=cos J I — j 



(27) 



The P.C.C. is moved backward. 

d. Assume that the radius of one curve is to be altered with- 
out changing either tangent. Assume conditions as in Fig. 29. 

For the diagrammatic solution 
assume that i?2 is to be increased 
by O2S. Then, since /?2' must 
pass through 0^ and extend be- 
yond Oi a distance O^S, the 
locus of the new center must lie 
on the arc drawn about 0^ as 
center and with OS as radius. 
The locus of O2 is also given 
by a line Oi^p parallel to BV 
and at a distance of i?2' (equal 
to S . . . P.C.C.) from it. The 
new center is therefore at the 
intersection O2'. An arc with ra- 
dius 7^2' ^^'ill therefore be tangent 
at B^ and tangent to the old 
Draw 0^n^ perpendicular to O2B, 



\syi 




Fig. 29. 



curve produced at new P.C.C. 



42 RAILROAD CONSTRUCTION. § 40. 

With O2 as center draw the arc O^m, and with O2' as center draw 
the arc 0{m\ mB=m'B' =R^, 

,'. mn=m'n^ =(R2 —Ri) vers J2' = (^2 — ^1) vers ij- 

.-. vers i/ = ^^^=|i^ vers i, (28) 

0{n = {R2—Ri) sin ijJ 
Oin' = (7?/-i?i)sin J/. 
BB' = 0X -Oin = (R2' -R,) sin J/-(7?2-^i) sin J2. (29, 

This problem may be further modified by assuming that the 
radius of the curve is decreased rather than increased, or that 
the smaller radius follows the larger. The solution is similar 
and is suggested as a profitable exercise. 

It might also be assumed that, instead of making a given 
change in the radius R2, a given change BB^ is to be made. Jj' 
and R2' are required. Eliminate /?2' from Eqs. 28 and 29 
and solve the resulting equation for J2. Then determine 7?2' 
by a suitable inversion of either Eq. 28 or 29. 

As in § § 32 and 33, the above problems are but a few, although 
perhaps the most common, of the problems the engineer may 
meet with in compound curves. All of the ordinary problems 
may be solved by these and similar methods. 

40. Problems, a. Assume that the two tangents of a com- 
pound curve are to be 348 feet and 624 feet, and that Ji = 22° 16' 
and ^2=28° 20'. Required the radii. 

[Ans. 7^1 = 326.92; 7^2 = 1574.85.] 

b. A line crosses a valley by a compound curve which is first 
a 6° curve for 46° 30' and then a 9° 30' curve for 84° 16'. It is 
afterward decided that the last tangent should be 6 feet farther 
up the hill. What are the required changes? [Note. The 
second tangent is evidently moved outward. The solution cor- 
responds to that in the first part of § 39, c. The P.C.C. is 
moved forward 16.39 feet. If it is desired to know how far the 
P.T. is moved in the direction of the tangent (i.e., the projection 
of BB\ Fig. 28, on F'^'), it may be found by observing that it 
is equal to nn' = (R 2— Ri) {sin Jj— sin J/). In this case it equals 
0.65 foot, which is very small because J^ is nearly 90°. The 
value of J2 (46° 30') is not used, since the solution is independent 
of the value of J 2- The student should learn to recognize 



§ 41. ALIGNMENT. 43 

which quantities are mutually related and therefore essential 
to a solution, and which are independent and non-essential. J 

TRANSITION CURVES. 

41. Superelevation of the outer rail on curves. When a mass 
is moved in a circular path it requires a centripetal force to keep 
it moving in that path. By the principles of mechanics we 
know that this force equals Gv^^gR, in which G is the weight, 
V the velocity in feet per second, g the accelerat ion of gravity in 
feet per second in a second, and R the radius of curvature. 
If the two rails of a curved track were laid on a level (trans- 
versely), this centripetal force could only be furnished by the 
pressure of the wheel-flanges against the rails. As this is very 
objectionable, the outer rail is elevated so that the reaction of 
the rails against the wheels shall 
contain a horizontal component 
equal to the required centripetal 
force. In Fig. 30, if ob represents 
the reaction, oc will represent the 

weight G, and ao will represent the — — --P "^^il^^^ — ^ — -^ 

required centripetal force. From •' \^ _^|_ --'l^' 

similar triangles we may write "m^"^ V ^ — 

sn : sm :: ao : oc. Call g = 32.17. \ \ '- 

Call R=o730-^D, which is suffi- 
ciently accurate for this purpose (see ^^* 
§ 19). Call r =5280F-3600, in which T' is the velocity in miles 
per hour, mn is the distance between rail centers, which, for 
an 80-lb. rail and standard gauge, is 4.916 feet sm is slightly 
less than this. As an average value we may call it 4.900, which 
is its exact value when the superelevation is 4i inches. Calling 
sn=e, we have 

e=sm—=4: 9— ^ 4.9X5280^721) 



oc ' gR G 32.17X36002X5730* 
e = . 0000572 F^Z) (30) 

It should be noticed that, according to this formula, the re- 
quired superelevation varies as the square of the velocity, which 
means that a change of velocity of only 10% would call for a 
change of superelevation of 21%. Since the velocities of trains 
over any road are extremely variable, it is impossible to adopt 



44 



RAILROAD CONSTRUCTIOX. 



42. 



any superelevation which will fit all velocities even approx- 
imately. The above fact also shows why any over-iefinement 
in the calculations is useless and why the above approximations, 
which are really small, are amply justifiable. For example, the 
above formula contains the approximation that R = d730^D. 
In the extreme case of a 10° curve the error involved would be 
about 1%. A change of about J of 1% in the velocity, or say 
from 40 to 40.2 miles per hour, would mean as much. The error 
in e due to the assumed constant value of srn is never more than 
a very small fraction of 1%. The rail-laying is not done closer 
than this The following tabular form is based on Eq. (30) : 

SUPERELEVATION OF THE OUTER RAIL (iN FEET) FOR VARIOUS 
VELOCITIES AND DEGREES OF CURVATURE. 



Velocity in 
Miles per 








Degree of Cun 


re. 








Hour. 


1° 


2° 


3° 


4^ 


5° 


6° 


7° 


8° 


9° 


10° 


30 


.05 
.09 
.14 
.20 


.10 
.18 
.29 
.41 


.15 
.27 
.43 


.20 
.37 


.26 
.46 


.31 


.36 


.41 


.46 


1.51 


40 


.86 


.64 


.73 


.82 




50 


4 -^^ 

.82 


"TT 




60 


1-62- 





42. Practical rules for superelevation. A much used rule for 
superelevation is to '' elevate one half an inch for each degree of 
curvature.^' The rule is rational in that e in Eq. 30 varies 
directly as D. The above rule therefore agrees with Eq. 30 
when T^ is about 27 miles per hour. However applicable the 
rule may have been in the days of low velocities, the elevation 
thus computed is too small now. The rule to elevate one inch 
for each degree of curvature is also used and is precisely similar 
in its nature to the above rule. It agrees with Eq. 30 when 
the velocity is about 38 miles per hour, which is more nearly 
the average speed of trains. 

Another (and better) rule is to ^'elevate for the speed of the 
fastest trains." This rule is further justified by the fact that a 
four-wheeled truck, having two parallel axles, will always tend 
to run to the outer rail and will require considerable flange pres- 
sure to guide it along the curve. The effect of an excess of super- 
elevation on the slower trains will only be to relieve this flange 
pressure somewhat. This rule is coupled with the limitation 



§42. 



ALIGNMENT. 



45 



that the elevation should never exceed a limit of six inches — 
sometimes eight inches. This limitation implies that locomo- 
tive engineers must reduce the speed of fast trains aroimd sharp 
curves until the speed does not exceed that for which the actual 
superelevation used is suitable. The heavy line in the tabular 
form (§ 41) shows the six-inch limitation. 

Some roads furnish their track foremen with a list of the super- 
elevations to be used on each curve in their sections. This 
method has the advantage that each location ma}^ be separately 
studied, and the proper velocity, as affected by local conditions 
{c.g.y proximity to a stopping-place for all trains), may be de- 
termined and applied. 

Another method is to allow the foremen to determine the 
superelevation for each curve by a simple measurement taken 
at the curve. The rule is developed as follows: By an inversion 
of Eq. 19 we have 

x = chord^^8R (31) 

Putting X equal to e In Eq. 30 and solving for '^chord/^ we 
have 

chord ^ = .Q000572V'L 7 

= 2.621T'2. 
chord = \.(j2V (32) 

To apply the rule, assume that 50 miles per hour is fixed as 
the velocity from which the superelevation is to be computed. 
Then 1.627 = 1.62X50 = 81 feet, Avhich is the distance given to 
the trackmen. Stretch a tape (or even a string) with a length 
of 81 feet between two points on the inside head of the outer rail 
or the outer head of the inner rail. The ordinate at the middle 
point then equals the superelevation. The values of this chord 
length for varying velocities are given in the accompanying 
tabular form. 



Velocity in miles per hour. . 
Chord length in feet 



20 
32.4 



30 
48.6 



35 
56.7 



40 45 
64.872.9 



50 

81.0 



55 60 

89.1 97.2 



The following tabular form shows the standard (at one time) 
on the N. Y., N. H. & H. R. R. It should be noted that the 
elevations do not increase proportionately with the radius, and 
that they are higher for descending grades than for le^el or 



46 



RAILROAD COXSTRUCTION. 



§43. 



ascending grades. This is on the basis that the velocity on curves 
and on ascending grades will be less than on descending grades. 
For example, the superelevation for a 0° 30' curve on a de- 
scending grade corresponds to a velocity of about 54 miles per 
hour, while for a 4° curve on a level or ascending grade the super- 
elevation corresponds to a velocity of only about 38 miles per 
hour. 



TABLE OF THE SUPERELEVATION OF THE OUTER RAIL ON CURVES^ 
N. Y., N. H. & H. R. R. 



Degree of 


Level or as- 


Descending 


curve. 


cending grade. 


grade. 




inches. 


inches. 


0° 30' 


Oi 


1 


1 00 


U 


U 


1 15 


li 


2 


1 30 


2 


2\ 


1 45 


21 


2i 


2 00 


2| 


2i 


2 15 


2i 


3 


2 30 


21 


3i 


2 45 


3 


31 


3 00 


3i 


3f 


3 15 


3i 


3i 


3 30 


3* 


4 


3 45 


3| 


4i 


4 00 


4 


4i 



43. Transition from level to inclined track. On curves the 
track is inclined transversely; on tangents it is level. The tran- 
sition from one condition to the other must be made gradually. 
If there is no transition curve, there must be either inclined 
track on the tangent or insufficiently inclined track on the curve 
or both. Sometimes the full superelevation is continued through 
the total length of the curve and the " run-off '^ (having a length 
of 100 to 400 feet) is located entirely on the tangents at each 
end. In other practice it is located partly on the tangent and 
partly on the curve. Whatever the method, the superelevation 
is correct at only one point of the run-off. At all other points 
it is too great or too small. This (and other causes) produces 
objectionable lurches and resistances when entering and leav- 
ing curves. The object of transition curves is to obviate these 
resistances. 

On the lichigh Valley R, R, the run-off is made in the form 
of a reversed vertical curve, as shown in the accompanying 
figure. According to this system the length of run-off varies 



§44. 



ALIGNMENT. 



47 



from 120 feet, for a superelevation of one inch, to 450 feet, 
for a superelevation of ten inches. Such a superelevation 
as ten inches is very unusual practice, but is successfully 
operated on that road. The curve is concave upward for two- 
thirds of its length and then reverses so that it is convex upward. 

TABLE FOR RUN-OFF OF ELEVATION OF OUTER RAIL OF CURVES. 
Drop in inches for each 30-foot rail commencing at theoretical point of curve. 



£.2 


V 


r 


r 


V i 


// 3// 


V 


V 


ir 


IV 


IF 


V 


r 


V 


r 


V 


r 


i 

V 


.*/ r 


^s'' 





W^ 




































1 




H 


V 




30 


30 




























30 




30 




120 


9// 




30 
30 
30 
30 
30 






. 30 
. 30 


30 












30 
30 


30 
30 


30 
30 
30 

30 


30 

30 
30 
30 


30 
30 


30 
30 
30 


• • • 


30 
30 




150 


3" 






180 


4" 




30 . 
30 . 
30 , 


30:. .. 


?A0 


5" 


30 








30 
30 


30 
30 




270 


6" 




30 


30 






300 


7// 




30 




30 




30 




30 




30 


30 




30 


30 




30 


30 




30 




330 


S" 




30 




30 


30 






30 


30 


30 


30 




30 


30 




30 




30 




30 


360 


9^^ 


30 






30 


, 30 




30 


30 




30 


30 


30 


30 


30 


30 


30 




30 




30 


420 


10'' 


30 




30 


. . 3 


. . 




30 


30 


30 


30 


30 


30 


30 


30 


30 


30 




30 




30 


450 




The figure (and also the lower line of the tabulated form) 
shows the drop for each thirty-foot rail length. For shorter 
lengths of run-off, the drop for each 30 feet is shown by the cor- 
responding lines in the tabular form. Note in each horizontal 
line that the sum of the drops, under which 30 is found, equals 
the total superelevation as found in the first column. For 
example, for 4 inches superelevation, length of curve 240 feet, 
the successive drops are Y', ¥\ y\ y% f", J'', i", and J" 
whose sum is 4 inches. Possibly the more convenient form 
would be to indicate for each 30-foot point the actual super- 
elevation of the outer rail, which would be for the above case 
(running from the tangent to the curve) J", |", J", IJ", 2|", 
. 3i", 3r', 4". 

44. Fundamental principle of transition curves. If a curve 



48 RAILROAD CONSTRUCTION. § 45. 

has variable curvature, beginning at the tangent with a curve 
of infinite radius, and the curvature gradually sharpens until it 
equals the curvature of the required simple curve and there 
becomes tangent to it, the superelevation of such a transition 
curve may begin at zero at the tangent, gradually increase to 
the required superelevation for the simple curve, and yet have 
at every point the superelevation required by the curvature at 
that point. Since in Eq. (30) e is directly proportional to D, 
the required curve must be one in which the degree of curve 
increases directly as the distance along the curve. The mathe- 
matical development of such a curve is quite complicated. It 
has, how^ever, been developed, and tables have been computed for 
its use, by Prof. C. L. Crandall. The following method has the 
advantage of great simplicity, while its agreement with the true 
transition curve is as close as need be, as will be shoTVTi. 

45. Multiform compound curves. If the transition curve com- 
mences with a very flat cur^e and at regular even chord lengths 
compounds into a curve of sharper curvature until the desired 
curvature is reached, the increase in curvature at each chord 
point being uniform, it is plain that such a curve is a close ap- 
proximation to the true spiral, especially since the rails as laid 
will gradually change their curvature rather than maintain a 
uniform curvature throughout each chord length and then 
abruptly change the curvature at the chord points. Such a 
curve, as actually laid, will be a much closer approximation 
to the true curve than the multiform compound curve by which 
it is set out. There will actually be a gradual increase in curva- 
ture w^hich increases directly as the length of the curve. 

46. Required length of spiral. The required length of spiral 
evidently depends on the amount of superelevation to be gained, 
and also depends somewhat on the speed. If the spiral is laid 
off in 25-foot chord lengths, with the first chord subtending a 1^. 
curve, the second a 2° curve, etc., the fifth chord will subtend 
a 5° curve, and the increase from this last chord to a 6° curve 
is the same as the uniform increase of curvature between the 
chords. The same spiral extended would run on to a 12° curve 
in (12 — 1)25=275 feet. The last chord of a spiral should have 
a smaller degree of curvature than the simple curve to w^hich it 
is joined. If the curves are very sharp, such as are used in street 
work and even in suburban trolley w^ork, an increase in degree 
of curvature of 1° Der 25 feet will not be sufficiently rapid, as 



§47. 



ALIGNMENT. 



49 



such a rate would require too long curves, 2^, 10°, or even 20° 
increase per 25 feet may be necessary, but then the chords 
should be reduced to 5 feet. Such 
a rapid rate of increase is justified 
by the necessary reduction in 
speed. On the other hand, very 
high speed will make a lower rate 
of increase desirable, and there- 
fore a spiral whose degree of cur- 
vature increases only 0° 30' per 25 
leet may be used. Such a spiral 
would require a length of 375 feet 
to run on to an 8° curve, w^hich is 
inconveniently long, but it might 
be used to run on to a 4° curve, 
where its length would be only 175 
feet. Three spirals have been de- 
veloped in Table IV, each with 
chords of 25 feet, the rate of in- 
crease in the degree of curvature 
being 0° 30', 1° and 2° per chord. 
One of these will be suitable for 
any curvature found on ordinary 
steam-railroads. 

47. To find the ordinates of a 
i°-per-25-feet spiral. Since the 
first chord subtends a 1° curve, its central angle is 0° 15' and the 
angle aQV (Fig. 31) is 7' 30". The tangent at a makes an 
angle of 15' w4th VQ. The angle between the chord ba and the 
tangent at a is i(30')=15', and the angle 6a6"= K30') + 15' 
= 30'. Similarly 

the angle chc'' = |(45') + 30' + 15' = 67' 30" = 1° 07' 30", 
and the angle dcd" =2° 0'. 

The ordinate aa' =25 sin 7' 30", and 
Qa'=25 cos 7' 30". 
Qy = Qa'+a'y 

=25 (cos 7' 30" + cos 30'). 
6&'=6'6" + 66" 

=25 (sin 7' 30" + sin 30'). 
Similarly the ordinates of c, d^ etc., may be obtained. 




Fig. 31. 



50 



RAILROAD CONSTRUCTION. 



§48. 



48. To find the deflections from any point of the spiral. 
aQV = 7'S0'\ Tan hQV = hh' ■-- Qh' \ tan cQV ^cc' -rQc' ) etc. 
Thus we are enabled to find the deflection angles from the tan- 
gent at Q to any point of the spiral. 

The tangent to the curve at c (Fig. 32) makes an angle of 

V 




Fig. 32. 
1°30' with QV, or cmF = l°30'. Qcm = cmV-cQm, The 
value of cQm is known from previous work. The deflection 
from c to Q then becomes known. 

acm = cmV — cap = cmV —caq—qap. caq is the deflection an- 
gle to c from the tangent at a and will have been previously 
computed numerically. qap = 15\ acm therefore becomes 
known. 

&cm = i of 45' =22' 30''; 
dcn = ^ of 60' =30'. 



§49. 



ALIGNMENT. 



51 



ecn=ecd'' —ncd'\ ncd^^=^cmV, tan ecd" =^{ee' —d"d'^-^-c'e\ all 
cf which are known from the previous work. 

By this method the deflections from the tangent at any point 




Fio. 33. 

of the curve to any other point are determinable. These values 
are compiled in Table IV The corresponding values of these 
angles when the increase in the degree of curvature per chord 
length is 30', and when it is 2°, are also given in Table IV. 

49. Connection of spiral with circular curve and with tangent. 
See Fig. 33.* Let AV and Z?F be the tangents to be connected 



♦ The student should at once appreciate the fact of the necessary distor- 
tion of the figure. The distance MM' in Fig. 33 is perhaps 100 times its real 
proportional value. 



52 RAILROAD CONSTRUCTION. § 49. 

by a D° curve, having a suitable spiral at each end. If no 
spirals were to be used, the problem would be solved as in simple 
curves giving the curve AMB Introducing the spiral has the 
effect of throwing the curve away from the vertex a distance 
MM^ and reducing the central angle of the D^ curve by 2(j>. 
Continuing the curve beyond Z and Z' to A' and B', we will 
have AA' =BB' = MAr, ZK = the x ordinate and is therefore 
known. Call MM' =m. A^N =x—R vers (j). Then 



7IT7IT/ A Af ^^ x-Rvers(^ .„^. 

m=MM'=AA^ = r-7 = rr-^' • • • • • (33) 

cos JJ cos iJ 



NA =AA' sin y = {x-R vers </>) tan J J. 
VQ = QK-KN^NA-hAV 

=y — R sin (j) -\-{x — R vers </>) tan hJ + R tan JJ 

= ?/ — J? sin 0-!-a: tan JJ + /? cos ^ tan JJ (34) 

When AW has already been computed, it may be more con- 
venient to write 



VQ=y-^R(tanU-smcj))+A'NtSiniJ (35) 

= i^exseciJH r-r t-t (3d) 

^ cos JJ COS JJ 



AQ=VQ-AV 

= 'y — R s\n ({)-{' (x — Rxers (j)) tSLU ^ J (37) 

A method of obtaining the necessary dimensions using tables, 
is given in § 53a. 

Example. To join two tangents making an angle of 34° 20' 
by a 5° 40' curve and suitable spirals. Use l°-per-25-feet spirals 
with five chords. Then = 3° 45', x = 2.999, iJ = 17° 10', and 
2/ = 124.942. 



I 



§50. 



ALIGNMENT. 



53 



[Eq. 33j 



[Eq. 36] 



[Eq. 35] 





R 


3.00497 




vers ^ 


7.33063 


2.166 




0.33560 


x= 2 999 






A'N= 0.833 




9.92064 




cos J J 


9.98021 


m=MM'=^A' = 0.872 


R 


9.94043 




3.00497 




Bxsec \d 


8.66863 


FAf = 47. 164 




1.67360 


m= 872 






FM' = 48.036 






^ = 124 . 942 nat. taii^ J = 


=.30891 




nat. sin ^ = 


=.06540 






.24351 


9.38651 




R 


3.00497 


246.314 


A'N 


2.39148 


[See above] 


9.92064 




tan J J 


9.48984 


0.257 


AAT 


9.41048 


rQ =371. 513 






R 


3.00497 




tan Ji 


9.48984 


312 471 


AF 


2.49481 



[Eq. 37] 



AQ= 59.042 

50. Field-work. When the spiral is designed during the 
original location, the tangent distance VQ should be computed 
and the point Q located. It is hardly necessar}^ to locate all of 
the points of the spiral until the track is to be laid. The ex- 
tremities should be located, and as there will usually be one 
and perhaps two full station points on the spiral, these should 
also be located. Z may be located by setting off QK=y and 
KZ=x, or else by the tabular deflection for Z from Q and the 
distance ZQ, which is the long chord. Setting up the instrument 
at Z and sighting back at Q with the proper deflection, the tan- 
gent at Z may be found and the circular curve located as usual, 
its central angle being J—2<f). A similar operation wuU locate 
Q' from 7/. 

To locate points on the spiral. Set up at Q, with the plates 



54 



RAILROAD CONSTRUCTION. 



§50. 



reading 0° when the telescope sights along VQ. Set off from 
Q the deflections given in Table IV for the instrument at Q, 
using a chord length of 25 feet^ the process being like the method 
for simple curves except that the deflections are irregular. If 
a full station-point occurs within the spiral, interpolate between 
the deflections for the adjacent spiral-points. For example, 
a spiral begins at Sta. 56 + 15. Sta. 57 comes 10 feet bej^ond 
the third spiral point. The deflection for the third point is 
35' 0"; for the fourth it is 56' 15". i| of the difference 
(21' 15") is 8' 30"; the deflection for Sta. 57 is therefore 43' 30". 
This method is not theoretically accurate, but the error is small. 
Arriving at Z, the forward alignment may be obtained by sight- 
ing back at Q (or at any other point) with the given deflection 




Fig. 34. 



for that point from the station occupied. Then when the plates 
read 0° the telescope will be tangent to the spiral and to the 
succeeding curve. All rear points should be checked from Z. 
If it is necessary to occupy an intermediate station, use the de- 
flections given for that station, orienting as just explained for Z, 



§ 51. ALIGNMENT. 55 

checking the back points and locating all forward points up to Z 
if possible. 

After the center curve has been located and Z' is reached, the 
other spiral must be located but in revenue order ^ i.e., the sharp 
curvature of the spiral is at Z' and the curvature decreases 
toward Q'. 

51. To replace a simple curve by a curve with spirals. This 
may be done by the method of § 49, but it involves shifting the 
whole track a distance m, which in the given example equals 
0.87 foot. Besides this the ti'ack is appreciably shortened, 
which would require rail-cutting But the track may be kept 
at practically the same length and the lateral deviation from the 
old track may be made very small by slightly sharpening the 
curvature of the old track, moving the new curve so that it is 
wholly or partially outs-ide of the old curve, the remainder of it 
with the spirals being inside of the old curve. It is found by 
experience that a decrease in radius of from 1% to 5% will 
answer the purpose The larger the central angle the less the 
change. The solution is as indicated in Fig. 34. 

0'N=R' cos <i>-ix, 
0'V=0'N sec iJ 

=R' cos (j) sec \d-\-x sec \d. 
m==MM'=MV-M'V 

= /eexsec JJ-(O'F-7?0 

=7? exsec ^J —R' cos ^ sec \d —x sec ^A-\-R\ . . . (38) 
AQ=QK-KN^NV-VA 

=y — R^ sin (j) + {R^ cos (f)-{ x) tan ^J—R tan J J 

=!/— E' sin ^ + i^'cos ^tan JJ-(/^— a;) tan JJ. . . (39) 



J 
The length of the old curve from Q to Q' = 2^4 Q + lOOy;- 

The length of the new curve from Q to Q' = 2L + 100-^7-^, 

in which L is the length of each spiral. 

Example. Suppose the old curve is a 7° 30' curve with a 
central angle of 38° 40'. As a trial, compute the relative length 
of a new 8° curve with spirals of seven chords, ^ = 7°0'; 
ii = 19°20'; R (for the 7° 30' curve) =764.489; R' (for the 
8° curve) =716.779; a;=7.628. 



56 RAILROAD CONSTRUCTION. § 52. 

[Eq. 38] 



[Eq. 39] 



45 . 687 




R 

exsec ^J 


2.88337 
8.77642 
1 65979 


«' = 716.779 
762.466 


753.953 . 

8.084 . 
762.037 

87.353 . 


R' 

cos 

sec^J 

X 

sec i J 

R' 

sin <p 

R' 

cos^ 
tan \J 




2.85538 
9.99675 
0.02521 

2.87734 




0.88241 
0.02521 
0.90762 


762.037 
m= 0.429 




2/ = 174.722 


2.85538 
9.08589 
1.94128 


249.606 


2.85538 
9.99675 
9.54512 

2 . 39725 


265.543 . 
352.896 


i2 = 764.489 
x= 7.628 
756.861 
tan ^A 






2.87901 
9.54512 

2.42415 


424.328 
352.896 







ylQ = 71.432 
The length of the old curve from Q to Q' is 



New curve: 100 



100^-100 ^^ - 


515.556 


2AQ = 2 X71.432 = 


142.864 


.-2^ j^^ 38.667 -14.000 3^^333 
U o.U 


658.420 


2L = 2 X 175 = 350.000 




658.333 


658.333 


Difference in length = 


= 0.087 



Considering that this difference may be divided among 22 
joints (using 30-foot rails) no rail-cutting would be necessary. 
If the difference is too large, a slight variation in the value of 
the new radius R' will reduce the difference as much as neces- 
sary. A truer comparison of the lengths would be found by 
comparing the lengths of the arcs. 

52. Application of transition curves to compound curves. 
Since compound curves are only employed when the location is 
limited by local conditions, the elements of the compound curve 
should be determined (as in §§38 and 39) regardless of the 



§52. 



ALIGNMENT. 



57 



transition curves, depending on the fact that the lateral shifting 
of the curve when transition curves are introduced is very small. 
If the limitations are very close, an estimated allowance may be 
made for them. 

Methods have been devised for inserting transition curves 
between the branches of a compound curve, but the device is 







Fig. 35. 



complicated and usually needless, since when the train is once 
on a curve the wheels press against the outer rail steadily and 
a change in curvature will not produce a serious jar even though 
the superelevation is temporarily a little more or less than it 
should be. 



58 RAILROAD CONSTRUCTION. § 53. 

If the easier curve of the compound curve is less than 3° or 
4°, there may be no need for a transition curve off from that 
branch. This problem then has two cases according as transition 
curves are used at both ends or at one end only. 

a. With transition curves at both ends. Adopting the method 
of § 49, calling ii = ^J, we may compute m^=MM^\ Similarly, 
calling J2 = ^^, we may compute m^=A/A/2'- I^ut AI/ and Af/ 
must be made to coincide. This may be done by moving the 
curve Z^Mi and its transition curve parallel to Q^V a distance 
M/M^j and ^the other curve parallel to QF a distance Mj'Mg. 
In the triangle M/MaMj', the angle at Afi' = 90° — J,, the angle 
at M2^ = 90° — J2y and the angle at 3/3 = J. 

mi n^ fur nj- /ir /Sin (90^ — ^2) / . cos ^2 1 

Then M/M3=M/M2' ^. — . — ^'= (mi— 7712)-. — ,-. 

^ "* sm J sm J 

y (40) 

CI- -1 1 Tir/ir ir /ir /Sin (90° — ii) . COS ij I 

Similarly Af2'M3 =Af /M 2 ^^ — 1 — - = (^h —^2J~- — 7. 

•^ ^ ^ sin J Sin J J 

b. With a transition curve on the sharper curve only. Com- 
pute m^=MM^' as before; then move the curve Z^M^^ parallel 
to Q^V a distance of 

M/M, = mi^^^ / (41) 

^ * ^ sin i ^ •^ 

The simple curve MA is moved parallel to VA a distance of 

MM,=mi^?^ (42) 

* ^ sin J 

If Ji and J2 are both small, M/M^ and MM^ may be more 
than mj, but the lateral deviation of the new curve from the old 
will always be less than m^. 

53. To replace a compound curve by a curve with spirals. 
The numerical illustration given below employs another method. 
We first solve for m^ for the sharper branch of the curve, plac- 
ing ii=i-^ in Eq. 38. A value for i^/ "^^y be found whose 
corresponding value of m^ will equal m^. Solving Eq. 38 for i^', 
we obtain 



c/ levers ^J — m cos^i--:r ^^^v 

cos ^ — cos ^J 



§ 53. ALIGNMENT. 59 

Substituting in this equation the known value of mj (=7^2) 
and caUing R'^^R^, R=R2, and J2 = iJ, solve for /?/, Obtain 
the value of AQ for each branch of the curve separately by Eq. 
39, and compare the lengths of the old and new lines. 

Example. Assume a compound curve with Z)i=8°, 1)2 =4°, 
Ji = 36^ and J. = 32°. Use l°-per-25-feet spirals; ^, = 7°0'; 
^2 = 1° 30'. Assume that the sharper curve is sharpened from 
8° 0' to 8° 12'. 

[Eq. 38] R, 2.85538 

exsec 36° 9.37303 

169.209 2.22842 

i«Ji' = 699.326 F^^^F^ 

ftfio .0;. i2/ 2.84468 

868-535 ^i 9.99675 

sec j\ 0- 09204 
857.970 2 ■ 93347 

3,, 0.88241 

sec Ji 0- 09204 

9.429 0.97445 

867.399 867.399 
mi- 1.136 



[Eq. 43] R2 3.15615 

vers 32° 9.18170 

217.700 2 . 33785 

m^ = l 136 . 05538 

^^0332° 9.92842 

0.963 9.98380 

a;2= 0.763 

1.726 1.726 j===- 

215.974 2.33440 

nat. cos <f> = .99966 
nat. cos J 2= .84805 

.15161 9.18073 

li;2'-1424.54 [4°1'22"] 3.15367 

[Eq.39] .. = 174.722 ^, |=^ 

sin 4>i 9 . 08589 

85.226 1.93057 

Ri' 2.84468 

cos <^, 9 . 99675 

tan i J [ii = 36°] 9.86126 

504.302 2 . 70269 

iJJi = 716.779 
a:i"° 7.628 

709.151 2.85074 

679:024 *^^*^ ^-^^i^ 

600.461 515.235 2.71200 

AQi- 78.563 600.461 



60 RAILROAD COXSTRUCTTON. § 53. 

[Eq. 39] Jt-l 3.15367 

2/2=- 74.994 sin <;62 8.41 1^ 1 

37.290 1.5715W 

R2' 3.15367 

cos j>2 9 . 99y85 

tan ^J(J2 = 32°) 9 . 79579 

889.843, 2.9-^931 

722=1432.69 
^2= 0.76 

1431.93 3.15592 
tan \d 9.79579 

894.770 2.95171 

964 . 837 932 . 060 
932 . 060 
AQ2^ "^2.111 
For the length of the old track we have : 

100^^' = 100^=450. 

Jo 32** 

100^' = 100 ^- = 800. 

AQ^ = 78.563 
AQ2 = 32 . 777 
1361.340 
For the length of the new track we have: 

^^^^^ = ^^^8^0"=" 353.659 

100^^=100^3= 758.140 

Spiral on 8° 12' curve 175.000 
" 4° 01' 22" " 75. 

Length of new track = 1361.799 

" " old " = 1361.340 

Exce.ss in length of new track = . 459 feet. 

Since the new track is slightly longer than the old, it shows 
that the new track runs too far outside the old track at the 
P.C.C. On the other hand the offset m is only 1.136. The 
maximum amount by which the new track comes inside of the 
old track at two points, presumably not far from Z' and Z, is 
very difficult to determine exactly. Since it is desirable that 
the maximum offsets (inside and outside) should be made as 
nearly equal as possible, this feature should not be sacrificed to 
an effort to make the two lines of precisely equal length so that 
the rails need not be cut. Therefore, if it is found that the offsets 
inside the old track are nearly equal to m (1.136), the above 
figures should stand. Otherwise m may be diminished (and the 
above excess in length of track diminished) by increasing R/ 
very slightly and making the necessary consequent changes. 



§ 53a. ALIGNMENT. 61 

53a. Use of Table IV. Prof. R. B. H. Begg, of Syracuse 
University, has submitted to the author a series of tables 
which will materially simplify the work of solving Equations 
(33) to (37), and which have been added to Table IV of pre- 
vious editions. Since these equations involve R and A (which 
may each have any values) in combination with several values 
of ^, it would require impracticably extensive tables to give 
precise values of the required dimensions for any possible com- 
bination of R, J, and ^. But the tables may be utilized hy 
interpolation with all necessary accuracy within their range. 
Rules for the use of the tables and for the field work are as 
follows: 

1. Find P. C. (point ^) as if no transition curve were to be 

used. 

2. Lay off the distance ^.V (part C of table) to N ; then offset 

the distance A'N (part B) to A', the new P. C; from N 
measure a distance NQ (part B) to Q. 

3. Set transit on A'; sight parallel to tangent and run in 

circular curve, setting Z from deflection and distance 
(part B)] or Z can be set by measuring ZK and QK 
(part B) from Q. 

4. Set transit on Q, sight along tangent and turn the deflection 

(part A) for each 25-foot station, for as many chord 
lengths as required; or the points may be located by 
measuring distances y along the tangent from Q and off- 
setting the corresponding distances x 

VERTICAL CURVES. 

54. Necessity for their use. Whenever there is a change in 
the rate of grade, it is necessary to eliminate the angle that 
would be formed at the point of change and to connect the two 
grades by a curve. This is especially necessary at a sag between 
two grades, since the shock caused by abruptly forcing an up- 
w^ard motion to a rapidly moving heavy train is very severe both 
to the track and to the rolling stock. The necessity for vertical 
curves was even greater in the days when link couplers were in 
universal use and the '' slack '^ in a long train was very great. 
Under such circumstances, when a train w^as moving down a 
heavy grade the cars w^ould crowd ahead against the engine. 
Reaching the sag, the engine w^ould begin to pull out, rapidly 
taking out the slack. Six inches of slack on each car w^ould 
amount to several feet on a long train, and the resulting jerk on 



62 RAILROAD CONSTRUCTION. § 54. 

the couplers, especially those near the rear of the train, has fre- 
quently resulted in broken couplers or even derailments. A 
vertical curve will practically elhninate this danger if the curve 
is made long enough, but the rapidly increasing adoption of 
close spring couplers and air-brakes, even for freight trains, is 
obviating the necessity for such very long curves. 

55. Required length. Theoretically the length should de- 
pend on the change in the rate of grade and on the length of the 
longest train on the road. A sharp change in the rate of grade 
requires a long curve; a long train requires a long curve; but 
since the longest trains are found on roads with, light grades and 
small changes of grade, the required length is thus somewhat 
equalized. It has been claimed that a total curve length equal 
to one-third of the train length for each tenth of a per cent of 
change of rate of grade will certainly prevent the rear of the 
train from crowding against the cars in front, but such a length 
is admittedly excessive. Half of this length is probably ample 
and one-fourth of it is probably safe. Therefore, we may say, 
taking the even fraction -^jj rather than yV, 

length of vertical curve = (length pf longest train) X (change 
of rate of grade in per cent). 

For example, assume a change of rate of grade of 2%; assume 
that the longest train will be about 720 feet. Then, by the 
above rule, the length of curve should be 720X2 = 1440 feet. 
Such rules are seldom if ever applied except in the most approx- 
imate way. On many roads a uniform length of only 400 feet 
is adopted for all vertical curves. The required length over 
a hump is certainly much less than that through a sag. Added 
length increases the amount of earthwork required both in cuts 
and fills, but the resulting saving in operating expenses will 
always justify a considerable increase. 

56. Form of curve. In Fig. 36 assume that A and C, equi- 



" t 

LEVEL LINE 



Fig. 36. 

distant from B, are the extremities of the vertical curve. Bisect 
AC at e; draw Be and bisect it at h. Bisect AB and BC at k 



§ 56. ALIGNMENT. 63 

and I. The line kl will pass through h. A parabola may be 
drawn with its vertex at h which will be tangent to AB and BC 
at A and C. It may readily be shown * from the properties of 
a parabola that if an ordinate be drawn at any point (as at n) 
w^e will have 

sn : eh (or KB) : '. An^ : Ae^ 

i^Arr? .... 

or sn=eh-^,—- (44) 

Ae^ 

In Fig. 36 the grades are necessarily exaggerated enormously. 
With the proportions found in practice we may assume that 
ordinates (such as mt, eB, etc.) are perpendicular to either 
grade, as may suit our convenience, without any appreciable 
error. In the numerical case given below, the variation of 
these ordinates from the vertical is 0° 07', while the effect of 
this variation on the calculations in this case (as in the most 
extreme cases) is absolutely inappreciable. It may easily be 
shown that the angle CA 5= half the algebraic difference of the 
rates of grade. Call the difference, expressed in per cent of 
grade, r; then CAB = ^r. Let Z=length (in '' stations" of 100 
feet) of the line AC, which is practically equal to the horizontal 
measurement. Since the angle CAB is one-half the total change 
of grade at B, it follows that Be = \lX Jr Therefore 

Bh = llr (44a) 

Since Bh \ov eh) are constant for any one curve, the correction 
sn at any point (see Eq. 44) equals a constant times Ani^. 

57. Numerical example. Assume that B is located at Sta. 
16 + 20; that the curve is to be 1200 feet long; that the grade 
of AB is -0.8%, and of 5C + 1.2%; also that the elevation 
of B above the datum plane is 162.6. Then the algebraic dif- 
ference of the grades, r, =1.2-(-0.8) =2.0; Z = 12. Bh = \lr 
= 1X12X2 = 3.0. A is at Sta. 10 + 20 and its elevation is 
162.6 + (6X0.8) =167.4; C is at Sta 22 + 20 and its elevation is 
162.6 + (6X1.2) =169.8. The elevation of Sta. 11 is found by 
adding sn to the elevation of s on the straight grade line. The 

constant {eh-^Ae^) equals in this ease 3.0-i-600^=^ifxfcTFTF' 
Therefore the curve elevations are 



* See note at foot of p. ")4. 



64 RAILROAD CONSTRUCTION. § 57. 

A, Sta. 10 + 20, 162. 6 + (6. 00X0. 8) =167.40 

11 167. 4-( .80X0.8) +i2tjW 802=166.81 

12 167.4-(1.80X0.8) +120W0 1802= 166.23 

13 167.4-(2.80X0.8) +i2o'o5u 2802= 165.81 

14 167.4-(3.80X0.8) +i2oVo<j 3802= 165.56 

15 167.4-(4.80X0.8) +i2o'ooff 4802=165.48 

16 167. 4 -(5. 80X0. 8) + 12 oW 580^=165.56 

B, 16 + 20,162.6 + 3.0 =165.60 

17 169.8-(5.20X1.2) +123555 5202= 165.81 

18 169.8-(4.20X1.2) +12^^000 4202=166.23 

19 169.8-(3.20X1.2) +120^055 3202=166.81 

20 169. 8 -(2. 20X1.2) +150W 2202= 167.56 

21 169.8-(1.20X1.2) +^055 1202=168.48 

22 169. 8-( .20X1.2) +120W 202=169.56 

C, 22 + 20, 162. 6 + (6. 00X1. 2) =169.80 



DEMONSTRATION OF EQ. 44. 

The general equation of a parabola passing through the point n (Fig. 36) 
may be written 

2/2 + 2/,,2 = 2pix + x,,h 

V- y n 

from which a-„ = — — + — x. 

" 2p 2p 

When X ^^ Xj^ j/ — y .^ and we have 

The general equation of a tangent passing through the point A may be 
written 

from which x = Xa 

When X =- a;^, 2/ = 2/g[= 2/,^]. and we have 



^5 


VuVa 
V 


- ^A. 






Va'^ 


yn'- 


^ynyj 


•^s 




2p 






(UA-yny 
2p 


Am' 
2p' 


2p 


yA' 

^A 


Ae' 

eh ' 




8n 


—rAm 







Tiiis proves the general proposi*^ion that if secants are drawn parallel to 
the axis of x, intersecting a parabola and a tangent to it, the intercepts be- 
tween the tangent and the parabola are proportional to the square of the 
distances (measured parallel to y) from the tangent point. 



CHAPTER in. 

EARTHWORK. 

FORM OF EXCAVATIONS AND EMBANKMENTS. 

58. Usual form of cross-section in cut or fill. The normal 
form of cross-section in cut is as shown in Fig. 37, in which 
e , . .g represents the natural surface of the ground, no matter 




how irregular; ah represents the position and width of the re- 
quired roadbed; ac and bd represent the "side slopes" which 
begin at a and h and which intersect the natural surface at such 




?2^ 



Fig. 38. 



points (c and d) as will be determined by the required slope 
angle (^). 

The normal section in fill is as shown in Fig. 38. The points 
c and d are likewise determined by the intersection of the re- 

65 



66 



RAILROAD CONSTRUCTION. 



§59. 



quired side slopes with the natural surface. In case the required 
roadbed (ab in Fig. 39) intersects the natural surface, both cut 




Fig. 39. 

and fill are required, and the points c and d are determined as 
before. Note that /? and /?' are not necessarily equal. Their 
proper values will be discussed later. 

59. Terminal pyramids and wedges. Fig. 40 illustrates the 
general form of cross-sections when there is a transition from 
cut to fill. a,. ,g represents the grade line of the road which 




Fig. 40. 

passes from cut to fill at d. sdt represents the surface profile. 
A cross-section taken at the point where either side of the road- 
bed first cuts the surface (the point m in this case) will usually 
be triangular if the ground is regular. A similar cross-section 
should be taken at o, where the other side of the roadbed cuts 
the surface. In general the earthwork of cut and fill terminates 



§ 60. EARTHWORK. 67 

in two pyramids. In Fig. 40 the pyramid vertices are at n 
and k, and the bases are Ihm and opq. The roadbed is generally 
wider in cut than in fill, and therefore the section Ihm and the 
altitude In are generally greater than the section opq and the 
altitude pk. When the line of intersection of the roadbed and 
natural surface (nodkm) becomes perpendicular to the axis of 
the roadbed (ag) the pyramids become wedges whose bases are 
the nearest convenient cross-sections. 

6o. Slopes, a. Cuttings. The required slopes for cuttings 
vary from perpendicular cuts, which may be used in hard rock 
which will not disintegrate b}^ exposure, to a slope of perhaps 
4 horizontal to 1 vertical in a soft material like quicksand or in 
a clayey soil which flows easily when saturated. For earthy 
materials a slope of 1 : 1 is the maxim.um allowable, and even 
this should only be used for firm material not easily affected by 
saturation. A slope of IJ horizontal to 1 vertical is a safer 
slope for average earthwork It is a frequent blunder that 
slopes in cuts are made too steep, and it results in excessive work 
in clearing out from the ditches the material that slides down, 
at a much higher cost per yard than it would have cost to take 
it out at first, to say nothing of the danger of accidents from 
possible landslides. 

b. Embankments. The slopes of an embankment vary from 
1 : 1 to 1.5 : 1. A rock fill will stand at 1 : 1, and if some care 
is taken to form the larger pieces on the outside into a rough 
dry wall, a much steeper slope can be allow^ed. This method is 
sometimes a necessity in steep side-hill work. Earthwork em- 
bankments generally require a slope of IJ to 1. If made 
steeper at first, it generally results in the edges giving way, re- 
quiring repairs until the ultimate slope is nearly or quite 1^ : 1. 
The difficulty of incorporating the added material with the old 
embankment and preventing its sliding off frequently makes 
these repairs disproportionately costly. 

6i. Compound sections. When the cut consists partly of 
earth and partly of rock, a compound cross-section must be 
made. If borings hav£ been made so that the contour of the 
rock surface is accurately known, then the true cross-section may 
be determined. The rock and earth should be calculated sepa- 
rately, and this will require an accurate knowledge of where the 
rock ''runs out" — a difl^cult matter when it must be deter- 



68 RAILROAD CONSTRUCTTOX. § 62. 

mined by boring. During construction the center part of the 
earth cut would be taken out first and the cut widened until a 
sufficient width of rock surface had been exposed so that the 
rock cut would have its proper width and ^de slopes. Then the 
earth slopes could be cut down at the proper angle. A^berm" 
of about three feet is usually left on the edges of the rock cut as 




Fig. 41. 

a margin of safety against a possible sliding of the earth slopes. 
After the work is done, the amount of excavation that has been 
made is readily computable, but accurate preliminary estimates 
are difficult. The area of the cross-section of earth in the figure 
must be determined by a method similar to that developed for 
borrow-pits (see § 89). 

62. Width of roadbed. Owing to the large and often dis- 
proportionate addition to A'olume of cut or fill caused by the 
addition of even one foot to the width of roadbed, there is a 
natural tendency to reduce the width until embankments become 
unsafe and cuts are too narrow for proper drainage. The cost 
of maintenance of roadbed is so largely dependent on the drain- 
age of the roadbed that there is true economy in making an 
ample allowance for it. The practice of some of the* leading 
railroads of the country in this respect is given in the following 
table, in which are also given some data belonging more properly 
to the subject of superstructure. 

It may be noted from the table that the average width 
for an earthwork cut, single track, is about 24.7 feet, with a 
minimum of 19 feet 2 inches. The widths of fills, single track, 
average over 18 feet, with numerous minimums of 16 feet. 
The widths for double track may be found by adding the distance 
between track centers, which is usually 13 feet. 



EARTHWORK. 



69 



1-(»-' ^ ® 



O fi 




























1 


iPfc 






^o. 1 


i! ^ «3-§ 




^CCCO COCOfO CCC^CO CO V 1 


«-^.»- c 




^^^ rH^rH rH t-I rH t-i (N | 






















r-i r-t r-* ,-( y-i ^ i-i T-i y-i T-ii-i T-i 


T-ll-l 













J* 




s 


lO iO»C»0»C^iC»0»OiOW5iO 


: uoic 






r-i ^' ^ ^' ^' rJ ,-H r4 •-< 1-i rH i-i' 




rt 




























0. 










! jO 






t-li-t rH _i-l 1- 




55 




3 


'• "lOOiO^iO "lO "lOiOiC ' 


••:.•• 1 






o 


rHH* ^.^.. .^Hf^ .^ 1 








T— 1 1-H 1-H T-t tH I— t i-H 1-H r-i 


" 










* i> 






— 








^' 




. CO 

•o ceo 


QOo<N 


coc> 










s 




•CO -t^co 


^ coco 


CO co- 


^ 












CO 


CO 
CO 


co 


; CO 




C3 


















3 
























•toco^ 


^M 


I "^ X 




2 








:xxx 


H..^^^ 


s rt 


c c^ 




o 




"3 




.(MC^Cq^ 


ooXx 


Xo «> 

<M CC% 


+ 








o 




■; + + + 

:Scoco 


co^S 


CO Tt^ 05 

CO CO (N -H 

^^ CO 












'. ^ S; 














-• 


O 


: o? ::^ 

•o coooQOco 


CO 




5: 


; K-co 






S 


(N 








b- 


; oiOi'"^ 


-i«i 














































o 








J X^^ ^ 


■ic 




^ 




'rl 






T^^iOCDtJ^ . 








o u . 


be 




_ij 




X 




2 -^ >.co 




3 


















s-vw-H^(N ^ 


CO 




^ <N(Nt1^ 








T-I 


T-I 






T-H(N 


























ti 




















t^ 




















&1 






1 




















.Sf2 






P 


























o 




















5^ 






CO 




















«yo^ 






?= 












i 




so 


icago, Burlington 
icago, Milwaukee 
C.,C.&St. Louis 
nois Central 


: > 

> 
if 






1 


> 


c 

^ d 
.2 








r ^jm :;:= fc, (u o3 o-- • o 


a> « 1 








<^ 


ooc 


"^ 


'■--! ^ 


h:i 


^ 


^ 


"^ 


^2; 


PlH 


P 


1 



70 RAILROAD CONSTRUCTION. § 63. 

63. Form of subgrade. The stability of the roadbed depends 
largely on preventing the ballast and subsoil from becoming 
saturated with water The ballast must be porous so that it 
will not retain w^ater, and the subsoil must be so constructed that 
it will readily drain off the rain-water that soaks through the 
ballast. This is accomplished by giving the subsoil a curved 
form, convex upward^ or a surface made up of two or three 
planes, the two outer planes having a slope of about 1 : 24 
(sometimes more and sometimes less, depending on the soil) 
and the middle plane, if three are used, being level. When a 
circular form is used, a crowning of 6 inches in a total width of 
17 or 18 feet is generally used. Occasionally the subgrade is 
made level, especially in rock-cuts, but if the subsoil is previously 
compressed by rolling, as required on the N. Y. C & H. R. R. R., 
or if the subsoil is drained by tile drains laid underneath the 
ditches, the necessity for slopes is not so great. Rock cuts are 
generally required to be excavated to one foot below subgrade 
and then filled up again to subgrade with the same material, if 
it is suitable. 

64. Ditches. ''The stability of the track depends upon the 
strength and permanence of the roadbed and structures upon 
which it rests; whatever will protect them from damage or pre- 
vent premature decay should be carefully observed. The worst 
enemy is w^ater, and the further it can be kept aw^ay from the 
track, or the sooner it can be diverted from it, the better the 
track will be protected. Cold is damaging only by reason of 
the water which it freezes; therefore the first and most impor- 
tant provision for good track is drainage." (Rules of the Road 
Department, Illinois Central R. R.) 

The form of ditch generally prescribed has a flat bottom 12'' 
to 24" wide and with sides having a minimum slope, except in 
rock-work, of 1 : 1, more generally 1.5 : 1 and sometimes 2:1. 
Sometimes the ditches are made V-shaped, w^hich is objection- 
able unless the slopes are low^ The best form is evidently that 
which w^ill cause the greatest flow^ for a given slope, and this 
. will evidently be the form in which the 
ratio of area to whetted perimeter is the 
largest. The semicircle fulfills this con- 

_ dition better than any other form, but the 

F[G. 42. 

nearly vertical sides would be difficult to 

maintain, (See Fig. 42.) A ditch, w4th a flat bottom and such 




§ 65. EARTHWORK. 71 

slopes as the soil requires, which approximates to the circular 
form will therefore be the best. 

When the flow will probably be large and at times rapid it 
will be advisable to pave the ditches with stone, especially if the 
soil is easily washed away. Six-inch tile drains, placed 2' under 
the ditches, are prescribed on some roads. (See Fig. 43.) No 
better method could be devised to insure a dry subsoil. The 
ditches through cuts should be led off at the end of the cut so 
that the adjacent embankment will not be injured. 

Wherever there is danger that the drainage from the land 
above a cut will drain down into the cut, a ditch should be made 
near the edge of the cut to intercept this drainage, and this 
ditch should be continued, and pa^ed if necessary, to a point 
where the outflow will be harmless Neglect of these simple 
and inexpensiA'e precautions frequently causes the soil to be 
loosened on the shoulders of the slopes during the progress of a 
heavy rain, and results in a landslide which will cost more to 
repair than the ditches which would have prevented it for all 
time. 

Ditches should be formed along the bases of embankments; 
they facilitate the drainage of water from the embankment, 
and may prevent a costly slip and disintegration of the em- 
bankment. 

65. Effect of sodding the slopes, etc. Engineers are unani- 
mously in favop of rounding off the shoulders and toes of em- 
bankments and slopes, sodding the slopes, paving the ditches, 
and providing tile drains for subsurface drainage, all to be put 
in during original construction. (See Fig. 43.) Some of the 
highest grade specifications call for the removal of the top layer 
of vegetable soil from cuts and from under proposed fills to 
some convenient place, from which it may be afterwards spread 
on the slopes, thus facilitating the formation of sod from grass- 
seed. But while engineers favor these measures and their 
economic value may be readily demonstrated, it is generally 
impossible to obtain the authorization of such specifications 
from railroad directors and promoters. The addition to the 
original cost of the roadbed is considerable, but is by no means 
as great as the capitalized value of the extra cost of mainte- 
nance resulting from the usual practice. Fig. 43 is a copy of 



RAILROAD CONSTRUCTION. 



§65. 



designs * presented at a convention of the Ameiican Society of 
Civil Engineers by Mr. D. J. Whittemore, Past President of 
the Society and Chief Engineer of the Chi., Mil. & St. Paul 




CUSTOMARY SECTION OF ROADBED ON EMBANKMENT. 

GRAVEL 




PROPOSED SECTION OF ROADBED ON EMBANKMENT. 

GRAVEU ♦ I t ^^ 




R. R. The ''customary sections '^ represent what is, with some 
variations of detail, the practice of many railroads. The " pro- 



♦ Trans. Am. Soc. Chdl Eng., Sept. 1894. 



§ 66. KARTHWORK. 73 

posed sections'^ elicited unanimous approval. They should be 
adopted when not prohibited by financial considerations. 

EARTHWORK SURVETS. 

66. Relation of actual volume to the numerical result. It 
should be realized at the outset that the accuracy of the result 
of computations of the volume of any given mass of earthwork 
has but little relation to the accuracy of the mere numerical 
work. The process of obtaining the volume consists of two 
distinct parts. In the first place it is assumed that the volume 
of the earthwork may be represented by a more or less com- 
plicated geometrical form, and then, secondly, the volume of 
such a geometrical form is computed. A desire for simplicity 
(or a frank w^illingness to accept approximate results) will often 
cause the cross-section men to assume that the volume may be 
represented by a very simple geometrical form w^hich is really 
only a very rough approximation to the true volume. In such 
a case, it is only a waste of time to compute the volume with 
minute numerical accuracy. One of the first lessons to be 
learned is that economy of time and effort requires that the 
accuracy of the numerical work should be kept proportional to 
the accuracy of the cross-sectioning work, and also that the 
accuracy of both should be proportional to the use to be made 
of the results. The subject is discussed further in § 94. 

67. Prismoids. To compute the volume of earthw^ork, it is 
necessary to assume that it has some geometric form whose vol- 
ume is readily determinable. The general method is to consider 
the volume as consisting of a series of prismoids ^ which are 
solids having parallel plane ends and bounded by surfaces which 
may be formed by lines moving continuously along the edges of 
the bases These surfaces may also be considered as the sur- 
faces generated by lines moving along the edges joining the cor- 
responding points of the bases, these edges being the directrices, 
and the lines being always parallel to either base, which is a 
plane director. The surfaces thus developed may or may not 
be planes. The volume of such a prismoid is readily determin- 
able (as explained in § 70 et seq.), while its definition is so very 
general that it may be applied to very rotigh ground. The 
^'two plane ends" are sections perpendicular to the axis of the 
road. The roadbed and side slopes (also plane) form three of 



74 



RAILROAD CONSTRUCTION. 



§68. 



the side surfaces. The only approximation lies in the degree of 
accuracy with which the plane (or warped) surfaces coincide with 
the actual surface of the ground between these two sections. 
This accuracy will depend (a) on the number of points which 
are taken in each cross-section and the accuracy with which the 
lines joining these points coincide wdth the actual cross-sections; 
(b) on the skill shown in selecting places for the cross-sections so 
that the warped surfaces shall coincide as nearly as possible with 
the surface of the ground. In fairly smooth country, cross- 
sections every 100 feet, placed at the even stations, are suf- 
ficiently accurate, and such a method simplifies the computations 
greatly; but in rough country cross-sections must be inter- 
polated as the surface demands. As will be explained later, 
carelessness or lack of judgment in cross-sectioning will introduce 
errors of such magnitude that all refinements in the computa- 
tions are utterly wasted. 

68. Cross-sectioning. The process of cross-sectioning con- 
sists in determining at an}^ place the intersection by a vertical 
plane of the prism of earth lying between the roadbed, the side 
slopes, and the natural surface. The intersection with the road- 




FiG. 44. 



bed and side slopes gives three straight lines. The intersection 
with the natural surface is in general an irregular line. On 
smooth regular ground or when approximate results are accept- 
able this line is assumed to be straight. According to the irreg- 



§ 69. EARTHWORK. 75 

ularity of the ground and the accuracy desired more and more 
'^intermediate points'^ are taken. 

The distance {d in Fig. 44) of the roadbed below (or above) 
the natural surface at the center is known or determined from 
the profile or by the computed establishment of the grade line. 
The distances out from the center of all " breaks '* are deter- 
mined with a tape. To determine the elevations for a cut, set 
up a level at any convenient point so that the line of sight is 
higher than an}^ point of the cross-section, and take a rod read- 
ing on the center point. This rod reading added to d gives the 
height of the instrument (H. I.) above the roadbed. Sub- 
tracting from H. I. the rod reading at any ''break" gives the 
height of that point above the roadbed (hi, ki, hr, etc.). This 
is true for all cases in excavation. For fill, the rod reading at 
center minus d equals the H. L, which may be positive or nega- 
tive. When negative, add to the "H. I." the rod readings of 
the intermediate points to get their depths below "grade"; 
when positive, subtract the "H. I." from the rod readings. 

The heights or depths of these intermediate points above or 
below grade need only be taken to the nearest tenth of a foot, 
and the distances out from the center will frequently be suffi- 
ciently exact when taken to the nearest foot. The roughness of 
the surface of farming land or woodland generally renders use- 
less any attempt to compute the volume with any greater accu- 
racy than these figures would imply unless the form of the ridges 
and hollows is especially well defined. The position of the slope- 
stake points is considered in the next section. Additional dis- 
cussion regarding cross-sectioning is found in § S2. 

69. Position of slope-stakes. The slope-stakes are set at the 
intersection of the required side slopes with the natural surface,^ 
which depends on the center cut or fill {d). The distance of 
the slope-stake from the center for the lower side is x = \h 
-I- s(cZ -h 2/) ; for the up-hill side it is x' = \h-{-s{d—y'). s is the 
''slope ratio" for the side slopes, the ratio of horizontal to ver 
tical. In the above equation both x and y are unknown. There- 
fore some position must be found by trial which will satisfy the 
equation. As a preliminary, the value of x for the point a = §& 
■\-sd, which is the value of x for level cross-sections. In the 
case of fills on sloping ground the value of x on the down-hill 
side is g^rea^er than this ; on ihQ wp-hill side it \s less. The differ- 
ence in distance is s times the difference of elevation. Take a 



76 RAILROAD CONSTRUCTION. ' § 69. 

numerical case corresponding with Fig. 45. The rod reading 
on c is 2.9; cZ = 4.2: therefore the telescope is 4.2—2.9 = 1.3 
helow grade. 6 = 1.5 : 1, 6 = 16. Hence for the point a (or for 
level ground) a: = ^X 16 + 1.5X4.2 = 14.3. At a distance out 
of 14.3 the ground is seen to be about 3 feet lower, which will 
not only require 1.5X3=4.5 more, but enough additional dis- 
tance so that the added distance shall be 1.5 times the additional 
drop. As a first trial the rod ma}^ be held at 24 feet out and a 
reading of, say, 8.3 is obtained. 8.3 + 1.3=9.6, the depth of 
the point below grade. The point on the slope line (n) which 
has this depth below grade is at a distance from the center 



Fig. 45. 

X = 8 + 1.5X9.6 =22.4. The point on the surface (s) having 
that depth is 24 feet out. Therefore the true point {m) is 
nearer the center. A second trial at 20.5 feet out gives a rod 
reading of, say, 7.1 or a depth of 8.4 below grade. This corre- 
sponds to a distance out of 20.6. Since the natural soil (espe- 
cially in farming lands or woods) is generally so rough that a 
difference of elevation of a tenth or so may be readily found by 
slightly varying the location of the rod (even though the dis- 
tance from the center is the same), it is useless to attempt too 
much refinement, and so in a case like the above the combina- 
tion of 8.4 below grade and 20.6 out from center may be taken 
to indicate the proper position of the slope-stake. This is 
usually indicated in the form of a fraction, the distance out being 
the denominator and the height above (or below) grade being 
the numerator; the fact of cut or fill may be indicated by C or F. 
Ordinarily a second trial will be sufficient to determine with 
sufficient accuracy the true position of the slope-stake. Ex- 
perienced men will frequent!}^ estimate the required distance 



§ 70. EARTHWORK. 77 

out to within a few tenths at the first trial. The left-hand pages 
of the note-book should have the station number, surface eleva- 
tion, grade elevation, center cut or fill, and rate of grade. The 
right-hand pages should be divided in the center and show the 
distances out and heights above grade of all points, as is illus- 
trated in § 84. The notes should read up the page, so that when 
looking ahead along the line the figures are in their proper 
relative position. The ''fractions'' farthest from the center 
line represent the slope-stake points. 

70. Setting slope-stakes by means of '' automatic'' slope-stake 
rods. The equipment consists of a specially graduated tape and 
a specially constructed rod. The tape may readily be prepared 
by marking on the back side of an ordinary 50-foot tape which is 
graduated to feet and tenths. Mark "0" at " ^6 " from the tape- 
ring. Then graduate from the zero backward, at true scale, to 
the ring. Mark off ''feet" and "tenths" on a scale propor- 
tionate to the slope ratio. For example, with the usual slope 
ratio of 1.5:1 each " foot " would measure 18 inches and each 
"tenth" in proportion. 

The rod, 10 feet long, is shod at each end and has an endless 
tape passing within the shoes at each end and over pulleys — to 
reduce friction. The tape should be graduated in feet and 
tenths, from to 20 feet — the and 20 coinciding. By moving 
the tape so that is at the bottom of the rod — or (practically) 
so that the 1-foot mark on the tape is one foot above the bottom 
of the shoe, an index mark may be placed on the back of the 
rod (say at 15 — on the tape) and this readily indicates when the 
tape is "set at zero." 

The method of use may best be explained from the figure and 
from the explicit rules as stated. The proof is given for two 
assumed positions of the level. 

(1) Set up the level so that it is higher than the "center" 
and (if possible) higher than both slope-stakes, but not more 
than a rod-length higher. On very steep ground this may be 
impossible and each slope-stake must be set by separate positions 
of the level. 

(2) Set the rod-tape at zero (i.e., so that the 15-foot mark 
on the hack is at the index mark) . 

(3) Hold the rod at the center-stake (J5) and note the read- 
ing (rii or 712). Consider n to be always plus; consider d to be 
plus for cut and minus for fill. 



78 



RAILROAD CONSTRUCTION. 



§70. 



(4) i2at^6 the tape on the /ace side of the rod (n + cO- AppHed 
literally (and algebraically), when the level is below the roadbed 
(only possible for fill), (?i + c?) = (712 + ( —d/)) =712 —d/. This being 
numerically negative, the tape is lowered (df—n^). With level 
at (1), for fill, (n + d) = (^1 + ( — c?/)) = (rii — d/) ; this being positive, 
the tape is raised. With level at (1), for cut, the tape is raised 
(ni + c?c). In every case the effect is the same as if the telescope 
were set at the elevation of the roadbed. 




Fig. 45a. 



(5) With the special distance-tape, so held that its zero is ^h 
from the center, carry the rod out until the rod reading equals 
the reading indicated by the tape. Since in cut the tape is 
raised (n + d), the zero of the rod-tape is always higher than the 
level (unless the rod is held at or below the elevation of the road- 
bed — which is only possible on side-hill work), and the reading 
at either slope-stake is necessarily negative. The reading for 
slope-stakes in fill is always positive. 

(6) Record the rod-tape reading as the numerator of a frac- 
tion and the actual distance out (read directly from the other 
side of the distance-tape) as the denominator of the fraction. 

Proof. Fill. Level at (i). Tape is raised (n^^—df). When 
rod is held at C/, the rod reading is +x, which =rf^ — {n^—df). 
But the reading on the back side of the distance-tape is also x. 

Fill. Level at (2). Tape is raised (n2—df), i.e., it is lowered 
{df—n^, ^Tien rod is held at C/, the rod reading is -\-Xj which 
similarly = r/j — (rij— c?/) = r/2 + {df—n^. Distance-tape as be- 
fore. 



§ 71. EARTHWORK. 79 

Cut Level at (i). Tape is raised (ni + c?c)- When rod is 
held at Cc the rod reading is— 2, which = rci — (ni + c/^c), i.e., 
2; = (rii + dci — Tci. The distance-tape will read z. 

Side-hill work. It is easily demonstrated that the method, 
when followed literally, may be applied to side-hill work, al- 
though there is considerable chance for confusion and error, 
when, as is usual, \h and the slope ratio are different for cut and 
for fill. 

The method appears complicated at first, but it becomes 
mechanical and a time-saver when thoroughly learned. The 
advantages are especially great when the ground is fairly level 
transversely, but decrease when the difference of elevation 
of the center and the slope-stake is more than the rod length. 
By setting the rod-tape ' ' at zero," the rod may always be used 
as an ordinary level rod and the regular method adopted, as in 
§ 69. Many engineers who have thoroughly tested these rods 
are enthusiastic in their praise as a time-saver. 



COMPUTATION OF VOLUME. 

71. Prismoidal formula. Let Fig. 46 represent a triangular 
prismoid. The two triangles forming the ends lie in parallel 
planes, but since the angles of one triangle are not equal to the 
corresponding angles of the other triangle, at least two of the 
surfaces must be warped. If a section, parallel to the bases, is 



Fig. 46. 

made at hi\y point at a distance x from one end, the area of the 
section will evidently be 



80 RAILROAD CONSTRUCTION. §71. 

The volume of a section of infinitesimal length will be Axdx, 
and the total volume of the prismoid will be * 



= J biK 



X + (h2-bi)h^^ + b,(h,-Jh)^^ 



:=-^A, + 4A,n + A,l (45) 

in which A^, A 2, and Am are the areas respectively of the two 
bases and of the middle section. Note that Am is not the mean 
of A J and A 2, although it does not necessarily differ .very greatly 
from it. 

The above proof is absolutely independent of the values, ab- 
solute or relative, of b^, 62, h^, or /12. For example, /12 "i^y be 
zero and the second base reduces to a line and the prismoid be- 
comes wedge-shaped; or &2 and /i2 niay both vanish, the second 
base becoming a point and the prismoid reduces to a pyramid. 
Since every prismoid (as defined in § 67) may be reduced to a 
combination of triangular prismoids, wedges, and pyramids, and ji 

* Students unfamiliar with the Integral Calculus may take for granted the 

fundamental formulae that / dx = x, that / xdx = ^x^, and that / x^dx^-^^x^; 

also that in integrating between the limits of I and (zero), the value of the 
integral may be found by simply substituting / for x after integration. 



§ 72. EARTHWORK. 81 

since the formula is true for any one of them individually, it is 
true for all collectively ; therefore it may be stated that * 

The volume of a prismoid equals one sixth of the perpendicular 
distance between the bases multiplied by the sum of the areas of 
the two bases plus four times the area of the middle section. 

While it is always possible to compute the volume of any 
prismoid by the above method, it becomes an extremely compli- 
cated and tedious operation to compute the true value of the 
middle section if the end sections are complicated in form. It 
therefore becomes a simpler operation to compute volumes by 
approximate formulae and apply, if necessary, a correction. 
The most common methods are as follows : 

72. Averaging end areas. The volume of the triangular 
prismoid (Fig. 46), computed by averaging end areas, is 

Z 

^[^^1^1 + 4^2^2]- Subtracting this from the true volume (as 

given in the equation above Eq. 45), we obtain the correction 

^[ih-b,)ih,-h,)] (46) 

This shows that if either the h's or 6's are equal, the correc- 
tion vanishes ; it also shows that if the bases are roughly similar 
and b varies roughly with h (which usually occurs, as will be 
seen later), the correction will be negative, w^hich means that the 
method of averaging end areas usually gives too large results. 

73. Middle areas. Sometimes the middle area is computed 
and the volume is assumed to be equal to the length tim,es the 

middle area. This w^ill equal — X ^ ^ X ^ ^ ^ . Subtracting 

Zi Ji Zi 

this from the true volume, we obtain the correction 

^^\-\-){K~K) (47) 

As before, the form of the correction shows that if either 
the /I's or 6's are equal, the correction vanishes-; also under the 
usual conditions, as before, the correction is positive and only 
one-half as large as by averaging end areas. Ordinarily the 
labor involved in the above method is no less than that of 
applying the exact prismoidal formula. 

* The student should note that the derivation of equation (45) does not 
complete the proof, but that the statements in the following paragraph are 
logically necessary for a general proof. 



82 



RAILROAD CONSTRUCTION. 



§74. 



74. Two-level ground. When approximate computations of 
earthwork are sufficiently exact the field-work may be materi- 
ally reduced by observing simply the center cut (or fill) and the 
natural slope a, measured with a clinometer. The area of such 
a section (see Fig. 48) equals 

\{a + d){xi + x,) ■ 



But 

from which 

Similarly, 
Substituting, 



2 * 

Xi tan p = a + d-\-xi tan a, 
a-\-d 



Xr = 



Area = (a + c?)2 



tan/? — tan a* 




a-\-d 


tan 


/? + tan a* 


^2 


tan ,5 



oh 
2 • 



(48) 



tan^ /? — tan^ a 

The values a, tan /?, tan^ /? are constant for all sections, so 
that it requires but little work to find the area of any section. 




Fig. 47. 



As this method of cross-sectioning implies considerable approxi- 
mation, it is generally a useless refinement to attempt to com- 




r 
Fig. 48. 



§ 75. EARTHWORK. 83 

pute the voiume with any greater accuracy than that obtained 
by averaging end areas. It may be noted that it may be easily 
proved that the correction to be appHed is of the same form as 
that found in § 72 and equals 

which reduces to 

6 (L tan2i3 — tan2a' tan^^ — tan2a"J' ■") 

When d" =d^ the correction vanishes. This shows that when 
the center heights are equal there is no correction — regardless 
of the slope. If the slope is uniform throughout, the form of the 
correction is simplified and is invariably negative. Under the 
usual conditions the correction is negative, i.e., the method 
generally gives too large results. 

75. Level sections. When the country is very level or when 
only approximate preliminary results are required, it is some- 
times assumed that the cross-sections are level. The method of 
level sections is capable of easy and rapid computation. The 
area may be written as 

{a-\-dys~ (50) 









^ I - 



Fig. 49. 

This also follows from Eq. 48 when a = and tan /?=— . 

s here represents the '' slope ratio," i.e.. the ratio of the horizontal 
projection of the slope to the vertical. A table is very readily 
formed gi^^ng the area in square feet of a section of given depth 
and for any given width of roadbed and ratio of side-slopes. 



84 RAILROAD CO^'STRUCTION. § 76. 

The area ma}^ also be readily determined (as illustrated in the 
following example) without the use of such a table; a table of 
squares Avill facilitate the work. Assuming the cross-sections 
at equal distances (=0 apart, the total approximate volume 
for any distance will be 

^[.lo + 2(.4i + .4,+ ... .4n-,)+^'ln]. . . . (51) 

The prismoidal correction may be directly derived from 
Eq. 46 as ^[2(a + d')s-2(a+d'')s][(a^\-d") -(a-]-d')l ^yh[ch 
reduces to 

-~(d^-d'r or --.^(d'-d-)\ . . (52) 



This may also be derived from Eq 49, since a = 0, tan a = 0, 
and ta.n ^ = 2a^h This correction is always negative, showing 
that the method of averaging end areas, when the sections ara 
level, always gives too large results The prismoidal correction 
for any one prismoid is therefore a constant times the square 
of a difference. The squares are always positive whether the 
differences are positive or negative. The correction therefore 
becomes 

-/g J-^(^'~^")'- • (53) 



76. Numerical example: level sections. Given the following 
center heights for the same number of consecutive stations lOG 
feet apart; width of roadbed 18 feet; slope IJ to 1. 

The products in the fifth column may be obtained very 
readily and with sufficient accuracy by the use of the slide-rule 
described in § 79. The products should be considered as 

(a + d)(a-\-d)^-' . In this problem s = lj, — = .6667. To apply 

the rule to the first case above, place 6667 on scale B over 89 
on scale A, then opposite 89 on scale B will be found 118.8 on 
scale A. The position of the decimal point will be evident from 
an approximate mental solution of the problem. 



§77. 



EARTH-VVORK. 



85 



Sta. 


Center 
Height. 


a-\-d 


(a+dy 


(a-\-d)2s 


Areas. 




d'^d" 


((f"-rf")2 


17 


2.9 


8.9 


79.21 


118.81 


118 


81 






18 


4.7 


10.7 


114.49 


171.741 


(343 


48 


1.8 


3.24 


19 


6.8 


12.8 


163.84 


245.76 . 
469.93 ^ 


y9- J491 
-^^-^ 939 


52 


2.1 


4.41 


20 


11.7 


17.7 


313.29 


86 


4.9 


24.01 


21 


4.2 


10.2 


104.04 


156.06 


312 


12 


7.5 


56.25 


22 


1.6 


7.6 


57.76 


86.64 


86 


64 


2.6 


6.76 



q6^6X18 
2 2 



= 54 

1752.43X100 
2X27 



3245 cub. yards 



2292.43 
10X54= 540 

1752.43 

= approx. vol. 



1 f\(\ vis 
Corr. = - ,' j^.;L X 94 . 67 = - 91 cub. yds. 



12X6X27 
3245 - 91 = 3154 cub. yds. 



= exact volume. 



94.67 



The above demonstration of the correction to be applied to 
the approximate volume, found by averaging end areas, is intro- 
duced mainly to give an idea of the amoimt of that correction. 
Absolutely level sections are practically unknown, and the error 
involved in assuming any given sections as truly level will 
ordinarily be greater than the computed correction. If greater 
accuracy is required, more points should be obtained in the 
cross-sectioning, which will generally show that the sections 
are not truly level. 

77. Equivalent sections. When sections are very irregular 
the following method may be used, especially if great accuracy 
is not required. The sections are plotted to scale and then a 
uniform slope line is obtained by stretching a thread so that the 
undulations are averaged and an equivalent section is obtained. 
The center depth (d) and the slope angle (a) of this line can be 
obtained from the drawing, but it is more convenient to measure 
the distances (xi and Xr) from the center. The area may then 
be obtained independent of the center depth as follows: Let 

b 



s = the slope ratio of the side slopes = cot /? = 



2a 



(See Fig. 50.) 



Then the 



Area = 



1 /xi-i-Xr\ , . . XrXr Xl Xl ah 



s 2 s 2 



XlXr 
S 



"2 



(54) 



86 



RAILROAD CONSTRUCTION. 



§78. 



The true volume, according to the prismoidal formula, of a 
length of the road measured in this way will be 



xi'xr _ ah 



/ xi' + xi'' x/- \-x/' 1 o^\ ^ 
\ 2 2 s 2 ) '^' 



If computed by averaging end areas, the approximate volume 
will be 



xi^x/ ah xi"xt" ah\ 
_"1 2~"^ s "2" J' 



Subtracting this result from the true volume, we obtain as the 
correction 

I 



Correction = —(x/' —a:/') (^/ —x/0. 



(55) 



This shows that if the side distances to either the right or 
left are equal at adjacent stations the correction is zero^ and 
also that if the difference is small the correction is also small 
and very probably within the limit of accuracy obtainable by 
that method of cross-sectioning. In fact, as has already been 
shown in the latter part of § 75, it will usually be a useless 




niiiiimiim . . 
"^1 -L, 



Fra. 50. 



refinement to compute the prismoidal correction when the 
method of cross- sectioning ia as rough and approximate as this 
method generally is. 

78. Equivalent level sections. These sloping " two-level '' 
sections are sometimes transformed into '^ level sections of equal 



§ 78. EARTHWORK. 87 

area/' and the volume computed by the method of level sections 
(§ 75). But the true volume of a prismoid with sloping ends 
does not agree with that of a prismoid with equivalent bases and 
level ends except under special conditions, and when this method 
is used a correction must be applied if accuracy is desired, 
although, as intimated before, the assumption that the sections 
have uniform slopes will frequently introduce greater inaccuracies 
than that of this method of computation. The following dem- 
onstration is therefore given to show the scope and limitations 
of the errors involved in this much used method. 

In Fig. 50, let dj^ be the center height which gives an equiva- 
lent level section. The area will equal (a-^diys — ^ , which 

must equal the area given in § 77, — -~^. s=^r~, 

or a-^d,=^^^^^ (56) 

To obtain c?, directly from notes, given in terms of d and a, 

we may substitute the values of xi and Xr given in § 74, which 

gives 

, , / , j\ tan /? a-^d ,^_ 

a+(^i = (a + rf) , ^ = . — . . (57) 

V tan^ /?— tan^ a \/l—s'^ tan^ a 

The true volume of the equivalent section may be repre- 
sented by 

From this there should be subtracted the volume of the 
"grade prism" under the roadbed to obtain the volume of the 
cut that would be actually excavated, but in the following com- 
parison, as well as in other similar comparisons elsewhere made, 
the volume of the grade prism invariably cancels out, and so for 
the sake of simplicity it will be disregarded. This expression 
for volume may be transposed to 

IsVxi'x/ /\/i7^' \/^?%^\ 2 , Xl^'Xr" "I 

6L s^ '^^\~2s~^^—2s—) +^^J- 



88 RAILROAD CONSTRUCTION. § 79. 

The true volume of the prismoid with sloping ends is (see § 77) 

The difference of the two volumes 



OS 



^^WxL'Xr"-\/xi"x/y. 



-2\/xi/Xr'x/W-Xrx/') 

^ (58) 



This shows that '^ equivalent level sections'' do not in 
general give the true volume, there being an exception when 
xi'xr" =^xi^'x/ . This condition is fulfilled when the slope is 
uniform, i.e., when a' = a" , When this is nearly so the error 
is evidently not large. On the other hand, if the slopes are in- 
clined in opposite directions the error may be very considerable, 
particularly if the angles of slope are also large. 

79. Three-level sections. The next method of cross-section- 
ing in the order of complexity, and therefore in the order of 




Fig. 51. 



accuracy, is the. method of three-level sections. The area of 
the section is \{a-\-d){vcT-^wi) — ^, which may be written 



§ 79, EARTHWORK. 8& 

^(a-{-d)w — ~', in which w=Wr + wi If the volume is com' 
puted by averaging end areas, it will equal 

~[{a + d')w'-'ah + {^a-{-d'')w''---ahl . . . (59) 

If we divide by 27 to reduce to cubic yards, we have, when 
Z = 100, 

Vol (,,,, . )=ff(a + c^0^^'-||a&4-|f(a + rf" V'-|f«?> 
For the next section 

Vol 0, . . ^) = ^(a + d")w''-f^ah + ma + d''')w'''-i^ab 
For a partial station length compute as usual and multiply 

result by — ^ — The prismoidal correction may be 

obtained by applying Eq 46 to each side in turn For the left 
side we have 

— [(« + ^0-(« + O](^/'-i^/'), which equals 

For the right side we have, similarly, 
~L(d'-d")(Wr"-Wr'). 

The total correction therefore equals 

-l^{d'-d")[(wi" + wr")-iwi' + wr')} 
= ~{d'-d"){w"-w'). 

Reduced to cubic yards, and with Z = 100, 

Pris. Corr.=||(<^'-0(^^"--'w^0. • . . (60) 

When this result is compared with that given in Eq. 55 there 
is an apparent inconsistency. If two-level ground is considered 
as but a special case of three -level ground, it would seem as if 



90 BAILROAD CONSTRUCTION. § 79. 

the same laws should apply If, in Eq. 55; x/=Xr', and a*/' 
is different from xl' , the equation i educes to zero; but in this 
case d' would also be different from c?"; and since xi'-\-Xr 
w^ould =w'j and xi" -\ Xt'^ =w'^ in Eq. 60, w" —w' would not 
equal zero and the correction Avould be some finite quantity and 
not zero. The explanation lies in the difference in the form 
and volume of the prismoids, according to the method of the 
formation of the warped surfaces If the surface is supposed to 
be generated by the locus of a line moving parallel to the ends 
as plane directors and along two sfraight lines lying in the side 
slopes, then xi^^^- will equal i(xi' -]-xi''), and a;r"^'<i- will equal 
i(x/ + x/'), but the profile of the center hne will not be straight 
and dmid. ^vill not equal i(d' + d''). On the other hand, if the 
surfaces be generated by two lines moving parallel to the ends 
as plane directors and along a straight center line and straight 
side lines lying in the slopes, a warped surface wdll be generated 
each side of the center line, which will have uniform slopes on 
each side of the center at the two ends and nowhere else This 
shows that when the upper surface of earthwork is warped (as 
it generally is), two-level ground should not be considered as a 
special case of three-level ground. This discussion, however, 
is only valuable to explain an apparent inconsistenc}^ and error. 
The method of two-level ground should only be used when 
such refinements as are here discussed are of no importance as 
affecting the accuracy. 

An example is given on the opposite page to illustrate the 
method of three-level sections. 

In the first column of yards 

210=fKa + cO^=MX7.3X31.1; 
507, 734, etc., are found similarly; 
595 =210 -61 -h 507 -61, • 

448 = xVo (507 -61 +734 -61); 
602 = iVo(734-61-h392-61); 

449 = 392-61+179-61. 

For the prismoidal correction, 

-20=ff(ci'-^'0(^''-'i^')=ff(2.6-8.1)(42.8-31.1) 
= ff(-5.5)( + 11.7). 

For the next line, -3 = /A[if(-2.6)( + S.7)], and similarly 
for the rest. The '' F^' in the columns of center heights, as well 



§79. 



EARTHWORK. 



91 



3d 



PUO 



I 









CO 

I 



OC' 

+ 





^ 


00 


»o 


1-1 


o 


^ 


1-1 




^ 


s 


CO 


'« 


CO 


00 


""t 


T— 1 


TJ^ 


+ 


t^ 


1-1 


lO 


- 


00 





• rJH • 00 
00 -1^2 

CO 1'^ "t 






k 


05 


fe. r- 


fe, 


^3 


fe. 


o 


fe. 


CC 




00, • 


04 




o 




00 




c^ 


• o 




(^ 




00 




c 


c^ 


iC CO 


g 


CO 


'it 


(N 


iO 



I-l Tt< 






i S- .2 



> 



I 55 



;5iw 



92 RAILROAD CONSTRUCTION. § 80. 

as in the columns of '^ right '^ and ''left/' are inserted to indicate 

fill for all those points. Cut would be indicated by " Cy 

25 
8o. Computation of products. The quantities ~{a-\-d)w 

^ i 

25 

and ^«& represent in each case the product of two variable 

terms and a constant. These products are sometimes obtained 
from tables which are calculated for all ordinary ranges of the 
variable terms as arguments. A similar table computed for 

25 

~r~{d' — d"){w" —w') will assist similarly in computing the 

prismoidal correction. Prof. Charles L. Crandall, of Cornell 
University, is believed to be the first to prepare such a set of 
tables, which were first published in 1886 in '' Tables for the 
Computation of Railway and Other Earthwork.' ' Another 
easy method of obtaining these products is by the use of a slide- 
rule. A slide-rule has been designed by the author to accom- 
pany this volume.* It is designed particularly for this special 
work, although it may be utilized for many other purposes for 
which slide-rules are valuable. To illustrate its use, suppose 
(a + c?) =28.2, and u' = 62.4; then 

25 , ,, 28.2X62.4 

Set 108 (which, being a constant of frequent use, is specially 
marked) on the sliding scale (J5) opposite 282 on the other scale 
(A), and then opposite 624 on scale B will be found 1629 on 
scale A, the 162 being read directly and the 9 read by estima- 
tion. Although strict rules may be followed for pointing off 
the final result, it only requires a very simple mental calculation 
to know that the result must be 1629 rather than 162.9 or 
16290. For products less than 1000 cubic yards the result 
may be read directly from the scale; for products between 1000 
and 5000 the result may be read directly to the nearest 10 

* The first edition of this book was octavo, and a pasteboard slide-rule, 
especially marked, accompanied each volume. Cutting down the size of 
^he pages to "pocket size " prevents the incorporation of the rule with the 
present edition. Any slide-rule with a logarithmic unit 22.V inches long will 
do equally well provided that the io8 mark is specially distinguished for 
ready use in computing the volume and that the 324 mark is similarly 
distinguished for use in computing the prismoidal correction. 



§ 81. EARTHWORK. 93 

yards, and the tenths of a division estimated. Between 5000 and 
10000 yards the result may be read directly to the nearest 20 
yards, and the fraction estimated; but prisms of such volume 
will never be found as simple triangular prisms — at least, an as- 
sumption that any mass of ground was as regular as this would 
probably involve more error than would occur from faulty esti- 
mation of fractional parts. Facilities for reading as high as 
10000 cubic yards would not have been put on the scale ex- 
cept for the necessity of finding such products as |f(9.lX9.5), 
for example. This product would be read off from the same 
part of the rule as |f (91X95). In the first case the product 
(80.0) could be read directly to the nearest .2 of a cubic yard, 
which is unnecessarily accurate. In the other case, the prod- 
uct (8004) could only be obtained by estimating /^ of a division. 
The computation for the prismoidal correction may be made 
similarly except that the divisor is 3.24 instead of 1.08. For 

example, ff(5.5Xll.7) =5^^^^. Set the 324 on scale B 

(also specially marked like 108) opposite 55 on scale .4, and 
proceed as before. 

8i. Five-level sections. Sometimes the elevations over each 
edge of the roadbed are observed when cross-sectioning. These 
are distinctively termed '^five-level sections." If the center, 
the slope-stakes, and one intermediate point on each side {not 
necessarily over the edge of the roadbed) are observed, it is 
termed an '^ irregular section." The field-work of cross-section- 
ing five-level sections is no less than for irregular sections with 
one intermediate point; the computations, although capable of 
peculiar treatment on account of the location of the intermediate 
point, are no easier, and in some respects more laborious; the 
cross-sections obtained w^ill not in general represent the actual 
cross-sections as truly as when there is perfect freedom in locat- 
ing the intermediate point; as it is generally inadvisable or un- 
necessary to employ five-level sections throughout the length of 
a road, the change from one method to another adds a possible 
element of inaccuracy and loses the advantage of uniformity of 
method, particularly in the notes and form of computations. 
On these accounts the method will not be further developed, 
except to note that this case, as well as any other, may be 
solved by dividing the whole prismoid into triangular prismoids, 
computing the volume by averaging end areas, and computing 



94 



RAILROAD CONSTRUCTION. 



§82. 



the prismoidal correction by adding the computed corrections 
for each elementary triar^ular prismoid. 

82. Irregular sections. In cross-sectioning irregular sections, 
the distance from the center and the elevation above ^' grade" 
of every ''break" in the cross-section must be observed. The 
area of the irregular section may be obtained by computing the 
area of the trapezoids {five, in Fig. 52 and subtracting the two 
external triangles. For Fig. 52 the area would be 



hi-\-ki, 



ki+_d 
2 

.kr^hr, 

+ -^, — ('^> 



d-^]'r . jr-\-kr . 



y^^ 2r 2; -2 V' 2)' 




Fig. 52. 



Expanding this and collecting terms, of which many will 

cancel, we obtain 



■40 



ArEA = — Xlkl-\-yi{d — hi)-\-Xrkr-{-lJr(jr—hr) 



•^Zr(d-kr)+ (hl+hr) 



} 



■ (61) 



An examination of this formula will show a perfect regu- 
larity in its formation which will enable one to write out a 
similar formula for any section, no matter how irregular or how 



§ 83. EARTHAVORK. ^ 95 

many points there are, without any of the preliminary work. 
The formula may be expressed in words as f ollow\s : 

Area equals one-half the sum of products obtained as follows : 

the distance to each slope-stake times the height above grade of 
the point next inside the slope-stake; 

the distance to each intermediate point in turn times the height of 
the point just inside minus the height of the point just outside; 

finally J one-half the loidth of the roadbed times the sum of the 
slope-stake heights. 

If one of the sides is perfectly regular from center to slope- 
stake, it is easy to show^ that the rule holds literally good. The 
^* point next inside the slope-stake" in this case is the center; 
the intermediate terms for that side vanish. The last term 
must always be used. The rule holds good for three-level sec- 
tions, in w^hich case there are three terms, which may be reduced 
to two. Since these two terms arc both variable quantities for 
each cross-section, the special method, given in § 78, in w^hich 

one term ( — j is a constant for all sections, is preferable. In 

the general method, each intermediate "break" adds another 
term. 

83. Volume of an irregular prismoid. This is obtained by 
computing first the approximate volume by ''averaging end 
areas " or by multiplying the length by the half sum of the end 
areas, as computed from Eq. (61). In other words, the Approx. 

volume = -^X^ (area' + area' ')• But since each area equals 

one-half the sum of products of width times height (see Eq. (61)) 
we may say that 

25 
Approx. volume = ^ (summation of width times height) . (62) 

the terms of width times height being like those found within 

the bracket of Eq. (61). 

As before, for partial station lengths, multiply the result by 

(length in feet -^ 100). There will be no constant subtractive 

25 
term, ^ ah, as in § 79. 

The correction to this approximate volume is found by 
considering that for the purpose of this correction ^nly the end 
sections are considered as ''three-level" sections and the cor- 
rection is computed by applying Eq. (60). 



96 



RAILROAD CONSTRUCTION. 



§84. 



84. Numerical example; irregular sections; volume with 

approximate prismoidal correction. Assume the earthwork 

notes as given below, where the roadbed is 18 feet wide in cut 

and the slope is 1^ to 1. Note that the stations read up the 

page and that when the surveyor is looking ahead along the 

line the several combinations of hetgnts and distances out have 

approximately the same relative position on the note book as 

they have on the ground. For example, beginning at the 

8 9c 
bottom line (St a. 16) the combination ^r^ means that the 

22.4 

extreme left-hand point of that section (the ''slope stake'O 

is 22.4 feet horizontally from the center and that it is 8.9 feet 

above the required roadbed. The cut (c) would he 8.9 feet 

to reach the roadbed, but of course the actual cutting is zero 

at the slope stake. The next point is 12.0 feet horizontally 

from the center and 7.0 feet above the roadbed. The cut at 

the center is 6.8 feet. The combinations of dimensions on the 

right-hand side are to be interpreted similarly. 



Sta. 

19 
18 
17 

+ 42 
16 



( cut 

Center K or 

/fill. 



0.6c 
2.3c 

7.6c 

10.2c 

6.8c 



3 . 6c 



22.4 



Left. 



14.4 






4.2c 


6.8c 


3.2c 


15.3 


8.4 


5.2 


8.2c 


10.2c 


8.0c 


21.3 


17.4 


6.1 


12. 2c 




12 6c 


27.3 




8.2 


8.9c 




7.6c 



12.0 



Right. 



0.1c 



4.1 



0.4c 



4.2 


9.6 




1.2c 




10.8 




4.2c 




15.3 


6.2c 


8.4c 


7.5 


21.6 


3.2c 


2.6c 



12.9 



The numerical computation is greatly facilitated by a sys- 
tematic form as given below. For Sta. 16, the first term is 
/'the distance to the left slope stake" (22.4) times "the height 
above grade of the point next inside '^ (the height being 7.6), 
and we place this pair of figures in the columns of "width" 
and "height." The "distance to the point next inside" is 
12.0 and the "height of the point just inside (6.8) minus the 



§85. 



EARTHWORK. 



97 



height of the point just outside" (8.9) equals ( — 2.1) and these 

25 

are the next pair of widths and heights. Taking ^ of the 

product of each pair of numbers we have the numbers in the 
first column of '^ yards." The sum of all these numbers in the 

42 

first and second groups multiplied by — -^ (that section being 

only 42 feet long) equals 378 cubic yards, the volume by averag- 
ing end areas. The determination of center heights and total 
widths and the application of Eq. (60), to obtain the approxi- 
mate prismoidal correction, is self-evident. 





VOLUME OF IRREGULAR 


PRISMOID, WITH APPROXIMATE 






PRISMOIDAL CORRECTION. 




Sta. 


W'th 


H'ght 


Yards. 


Cen. 
Height. 


Total 
width 


d'-d^' 


w"-w' 


Approx. 
pris.corr. 




22.4 


7.6 


158 




+ 6.8 


35.3 










12.0 


-2.1 


-23 














16 


12.9 
4.1 
9.0 


3.2 

4.2 

11.5 


40 
16 
96 
















27.3 


12.6 


319 




+ 10.2 


48.9 


-3.4 


+ 13.6 


-14 




8.2 


-2.0 


-15 














+ 42 


21.6 
7.5 


6.2 
1.8 


124 
13 
















9.0 


20.6 


172 


378 










(-6) 




21.3 


10.2 


201 




+ 7.6 


36.6 


+ 2.6 


-12.3 


-10 




17.4 


-0.2 


- 3 














17 


6.1 
15.3 


-2.6 
7.6 


-14 

107 
















9.0 


12.4 


103 


584 










(-6) 




15.3 


6.8 


95 




+ 2.3 


26.1 


+ 5.3 


-10.5 


-17 




8.4 


-1.0 


- 7 














18 


5.2 

10.8 


-4.5 
2.3 


-22 
23 
















9.0 


5.4 


45 


528 










(-17) 




14.4 


0.6 


8 




+ 0.6 


24.0 


+ 1.7 


-2.1 


-1 


19 


9.6 


0.1 


1 














4.2 


0.2 


1 
















9 


4.0 


33 


177 










(-1) 



-30 



Approx. volume =1667 
Approx. pris. corr. = —30 

Corrected volume ^ 1637 cubic yards 

85. Magnitude of the probable error of this method. In 
previous editions of this work, methods were given for com- 
puting the mathematically exact volume of a prismoid whose 
ends coincide with the '^irregular sections" as measured, and 



98 RAILROAD CONSTRUCTION. § 85. 

whose upper surfaces are assumed to coincide with the actual 
surface of the ground. As in the previous methods, the '^ap- 
proximate volume'' is computed by averaging end areas and 
then a correction is applied. If the end sections have the same 
number of intermediate points on each side, and if it can be as- 
sumed that the corresponding lines in each section are connected 
by plane or warped surfaces, which coincide with the surface of 
the ground, then the mathematically exact or 'Hrue'' correc- 
tion may be obtained by dividing the volume into elementary 
triangular prismoids, finding the correction for each and adding 
the results. Although such a method appears very complicated, 
it is readily possible to develop a law by means of which the 
true prismoidal correction may be written out (similarly to 
writing out the formula for the area, Eq. (61)) without any 
preliminary calculation. Such a law has a mathematical 
fascination, but it should be remembered that when the ground 
surface is so broken up that the cross-sections are ''irregular" 
it is in general correspondingly rough and irregular between 
the cross-sections, especially when those sections are 100 feet 
apart. It is also true that the cross-sections do not usually 
have the same number of intermediate points on corresponding 
sides of the center. In such a case, unless the actual form of 
the ground between the cross-sections is observed and measured, 
the. exact method cannot be used. An extra point in one cross- 
section implies an extra ridge (or hollow) which "runs out" 
or disappears by the time the adjoining section is reached. 
Theoretically a cross-section should be taken at the point where 
such a ridge or hollow runs out. In general this point will not 
be at an even 100-foot station. The attempt to compute the 
exact prismoidal correction usually gives merely a false appear- 
ance of extreme accuracy to the work which is not justified 
by the results. It should not be forgotten that it is readily 
possible to spend an amount of time on the surveying and 
computing which is worth more than the few cubic yards of 
earth which represents the additional accuracy of the more 
precise method. The accuracy of the office computation should 
be kept proportionate to the accuracy of the cross-sectioning 
in the field. The discussion of the magnitude of the prismoidal 
correction in §§ 72-79 shows that it is small except when the 
two ends of the prismoid are very dissimilar. The dissimiliarity 
between the two ends of a prismoid would be substantially the 



88. 



EARTHWORK. 



99 



same whether the ends were actually ''irregular*' or had ''three- 
level'' sections, which for each end had the same slope stakes 
and center heights as the irregular sections. Experience proves 
that the approximate prismoidal correction, computed by 
considering the ground as three-level, is so nearly equal to the 
true prismoidal correction that the difference is perhaps no 
greater than the probable difference between the true volume 
of earth and the volume of the geometrical prismoid which is 
assumed to represent that volume. The experienced surveyor 
will take his cross-sections at such places and so close together 
that the warped surfaces joining the sections w^ll lie very nearly 
in the surface or at least will so average the errors that they 
will substantially neutralize each other. 

86. Numerical illustration of the accuracy of the approxi- 
mate rule. The "true" prismoidal correction for the numerical 
case given in § 84 was computed by the method outlined above, 
and on the basis of certain figures as to the vanishing of the 
ridges and valleys found in one section and not found in the 
adjacent sections. The various quantities for the volumes 
between the cross-sections have been tabulated as shown. 





1 


2 


3 


4 


5 


6 


7 






^ tx 




di 


^.^% 


(N 


•-^ (-< ^ 


CO 






m 




a 

3 


m^ 


'^t 




^'"g 


Sect 


ons. 


« 2 2- 
o a) rt 

IN 


^2 


O 
> 


Approx. 

corr. on 

of three 

groun 




Approx 

compu 

from ce 

and SI 

heights 


3 


16 


.16 + 42 


378 


- 5 


373 


- 6 


-1 


396 


-23 


16 + 42. 


.17 


584 


- 3 


581 


- 6 


-3 


577 


+ 4 


17 


.18 


528 


-16 


512 


-17 


-1 


463 


+ 49 


IS 


.19 


177 


- 3 


174 


- 1 


+ 2 


147 


+ 27 




1667 


-27 


1640 


-30 


-3 


1583 


+ 57 



There has also been shown in the last two columns the error 
involved if the "intermediate points" had been ignored in 
the cross-sectioning. From the tabular form we may learn that 

1. The differences between the "true" and approximate 
corrections is so small that it is probably swallowed up by errors 
resulting from inaccurate cross-sectioning. 

2. The error which would have been involved in ignoring 
the intermediate points is so very large in comparison with 



100 RAILROAD CONSTRUCTION. 



W. 



the other corresponding errors that (although it proves nothing 
absolutely definite, being an individual case) the prohabilities 
of the relative error from these sources are clearly indicated. 

87. Cross-sectioning irregular sections. The slope stake 
should preferably be determined first, and then the *' breaks" 
between the slope stake and the center. When, as is usual, 
the ground is not even between the cross-section just taken 
and the section at the next 100-foot station, a point should be 
selected for a cross-section such that the lines to the previous 
section should coincide with the actual surface of the ground as 
closely as the accuracy of the work demands. § 94 gives 
a numerical illustration of the magnitude of some of these 
3rrors. Although it is possible for a skillful surveyor to so 
choose his cross-sections in rough and irregular ground that 
the positive and negative errors will nearly balance, it requires 
exceptional skill. Frequently the work may be simplified by 
computing separately the volume of a mound or pit, the 
existence of which has been ignored in the regular cross- 
sectioning. 

88. Side-hill work. When the natural slope cuts the roadbed 
there is a necessitv for both cut and fill at the same cross-section. 



Fig. 53. 

When this occurs the cross-sections of both cut and fill are often 
so nearly triangular that they may be considered as such without 
great error, and the volumes may be computed separately as 
triangular prismoids without adopting the more elaborate form 
of computation so necessary for complicated irregular sections. 
When the ground is too irregular for this the best plan is to 



§88. 



EARTHWORK. 



101 



follow the uniform system. In computing the cut, as in Fig. 53. 
the left side would be as usual; there would be a small center 
cut and an ordinate of zero at a short distance to the right of the 
center. Then, ignoring the fill, and applying Eq. 61 strictly, 
we have two terms for the left side, one for the right, and the 
term involving ^6, which will be ^bhi in this case, since hr=0, 
and the equation becomes 

Area, = i[xiki + yi(d— hi) -\-Xrd-\-ihhi], 

The area for fill may also be computed by a strict application 
of Eq. 61, but for Fig. 54 all distances for the left side are zero 
and the elevation for the first point out is zero, d also must be 





'h £% 


X^ 


k — --2/--H; 


^v 


r — r^^-^-^ 1 


-^ 


*S3 n 


s 
\ 

\ 




\ 

\ 
\ 


I y 



Fig. 54. 



considered as zero. FolloAving the rule, § 81, literally, the equa- 
tion becomes 

Area(FiU) =i[^r^v + !/r(o— /ir) +2r(o— A>) + J&(o+/ir)], 

which reduces to 

i[Xrkr—yrhr — Zrkr + hbhr]. 

(Note that Xr, hr, etc., have different significations and values 
in this and in the preceding paragraphs.) The ''terminal 
pyramids" illustrated in Fig. 40 are instances of side-hill work 
for very short distances. Since side-hill work always implies 
both cut and fill at the same cross-section, whenever either the 
cut or fill disappears and the earthwork becomes wholly cut or 
wholly fill, that point marks the end of the '' side-hill work," 
and a cross-section should be taken at this point. 



102 RAILROAD CONSTRUCTION. § 89. 

89. Borrow-pits. The cross-sections of borrow-pits will vary 
not only on account of the undulations of the surface of the 




Tig. 55. 

ground, but also on the sides, according to whether they are 
made by widening a convenient cut (as illustrated in Fig. 55) 
or simply by digging a pit. The sides should always be prop- 
erly sloped and the cutting made cleanly, so as to avoid un- 
sightly roughness. If the slope ratio on the right-hand side 
(Fig. 55) is s, the area of the triangle is ism'^. The area of the 
section is i[ug-\-(g-\-h)v + (h-hf)x + (j-{-k)ij + (k + 7n)z—sm^l If 
all the horizontal measurements were referred to one side as 
an origin, a formula similar to Eq. 61 could readily be devel- 
oped, but little or no advantage would be gained on account of 
any simplicity of computation. Since the exact volume of the 
earth borrowed is frequently necessary, the prismoidal correc- 
tion should be computed; and since such a section as Fig. 55 
does not even approximate to a three-level section, the method 
suggested in § 83 cannot be employed. It will then be neces- 
sary to employ the more exact method of dividing the volume 
into triangular prismoids and taking the summation of their 
corrections, found according to the general method of § 72. 

90. Correction for curvature. The volume of a solid, gen- 
erated by revolving a plane area about an axis lying in the 
plane but outside of the area, equals the product of the given 
area times the length of the path of the center of gravity of the 
area. If the centers of gravity of all cross-sections lie in the 
center of the road, where the length of the road is measured, 
there is absolutely no necessary correction for curvature. If all 
the cross-sections in any given length were exactly the same and 
therefore had the same eccentricity, the correction for curvature 
would be very readily computed according to the above prin- 
ciple. But when both the areas and the eccentricities vary 
from point to point, as is generally the case, a theoretically exact 



§ 90. EARTHWORK. 103 

solution is quite complex, both in its derivation and application. 
Suppose, for simplicity, a curved section of the road, of uniform 
cross-sections and with the center of gravity of every cross- 
section at the same distance e from the center line of the road 
The length of the path of the center of gravity will be to the 
length of the center line as R±e:R. Therefore we have 

R±e 
True vol.: nominal vol, ::R±e -R. .*. True voL=lA ^ for 

a volume of uniform area and eccentricity. For any other area 

R±e' 
and eccentricity we have, similarly. True vol/ =IA' — ^— . This 

shows that the effect of curvature is the same as increasing (or 

diminishing) the area by a quantity depending on the area and 

eccentricity, the increased (or diminished) area being found by 

R^e 
multiplying the actual area by the ratio — ^— . This being 

independent of the value of I, it is true for infinitesimal lengths. 
If the eccentricity is assumed to vary uniformly between two 
sections, the equivalent area of a cross-section located midway 

(«±^') 

between the two end cross-sections would be Am— 5 • 

Therefore the volume of a solid which, w^hen straight, w^ould be 
— (A' + 4A,n + A"), would then become 

TrueroZ.=^rA'(i2±60+4.4„,(i2±?^'WA''(i?±e'ol. 

Subtracting the nominal volume (the true volume when the 
prismoid is straight), the 



Correciion = ±^r(A' + 2A,n)e' + (2A^+A'0e'' 1. 



(63) 



Another demonstration of the same result is given by Prof. 
C. L. Crandall in his ^'Tables for the Computation of Railway 
and other Earthwork," in which is obtained by calculus methods 
the summation of elementary volumes having variable areas 
with variable eccentricities. The exact application of Eq. 63 
requires that A,^ be known, which requires laborious computa- 



104 



RAILROAD CONSTRUCTION. 



§91. 



tions, but no error worth considering is invoh ed if the equation 
is written approximately 



Curv. corr.=^{A'e' + A'^e''), 
2K 



(64) 



which is the equation generally used. The approximation con- 
sists in assuming that the difference between A' and A^i equals 
the difference between A wi and A" but with opposite sign. The 
error due to the approximation is always utterly insignificant. 

91. Eccentricity of the center of gravity. The determination 
of the true positions of the centers of gravity of a long series of 
irregular cross-sections would be a very laborious operation, 
but fortunately it is generally sufficiently accurate to consider 
the cross-sections as three -level ground, or, for side-hill work, to 




Fig. 56. 



be triangular, for the purpose of this correction.. The eccentricity 
of the cross-section of Fig. 56 (including the grade triangle) may 
be written 



(a-\-d)xi Xi 
2 ^" 



(a-\-d)XrXr 



{a-\-d)xj {a-]-d)Xr 



1 Xi^—Xr"^ 
3 Xi + Xr 



= 7(0:^-0:,.). 



(65) 



The side toward xi being considered positive in the above 
demonstration, if Xr>xi, e would be negative, i.e., the center 
of gravity would be on the right side. Therefore, for three-level 



§ 91. EARTHWORK. 105 

ground, the correction for curvature (see Eq. 64) may be written 
Correction = -^ [A\xi'-x/) + A"(a;/' -x/OJ- 
Since the approximate volume of the prismoid is 

in which V and 7" represent the number of cubic yards corre- 
sponding to the area at each station, we may write 

Corr. in cuh. yds. = ^[V\x/ -x/) + V'^x/' -x/'n • (66) 

It should be noted that the value of e, derived in Eq. 65, is 
the eccentricity of the whole area including the triangle under 
the roadbed. The eccentricity of the true area is greater than 
this and equals 

, true area + iah 



eX- 



true area 



The required quantity (A'e' of Eq. 64) equals true areaXe^ 
which equals {true area + ^ab) X e. Since the value of e is very 
simple, while the value of e^ would, in general, be a complex 
quantity, it is easier to use the simple value of Eq. 65 and add 
\ab to the area. Therefore, in the case of three-level ground 
the subtractive term |f a5 (§ 79) should not be subtracted in 
computing this correction. For irregular ground, when com- 
puted by the method given in §§82 and 83, which does not 
involve the grade triangle, a term ^^ah must be added at every 
station when computing the quantities F' and F" for Eq. 66. 

It should be noted that the factor 1-^-3/?, which is constant 
for the length of the curve, may be computed with all necessary 
accuracy and w^ithout resorting to tables by remembering that 

5730 



degree of curve' 



Since it is useless to attempt the computation of railroad 
earthwork closer than the nearest cubic yard, it will frequently 



106 RAILROAD CONSTRUCTION. § 91. 

be possible to write out all curvature corrections by a simple 

mental process upon a mere inspection of the computation sheet. 

Eq. 66 shows that the correction for each station is of the form 

y(xi—x ) 
\,r, • 3i^ is generally a large quantity — for a 6° curve 

it is 2865. (xi—Xr) is generally small. It may frequently be 
seen by inspection that the product V(xi—Xr) is roughly twice 
or three times 3R, or perhaps less than half of 3 J?, so that the 
corrective term for that station may be written 2, 3, or cubic 
yards, the fraction being disregarded. For much larger absolute 
amounts the correction must be computed with a correspondingly 
closer percentage of accuracy. 

The algebraic sign of the curvature correction is best deter- 
mined by noting that the center of gravity of the cross-section is 
on the right or left side of the center according as Xr is greater 
or less than xi, and that the correction is positive if the center of 
gravity is on the outside of the curve, and negative if on the 
inside. 

It is frequently found that xi is uniformly greater (or uni- 
formly less) than Xr throughout the length of the curve. Then 
the curvature correction for each station is uniformly positive or 
negative. But in irregular ground the center of gravity is apt 



Fig. 57. 

to be irregularly on the outside or on the inside of the curve, 
and the curvature correction will be correspondingly positive or 
negative. If the curve is to the right, the correction will be 
positive or negative according as (xi—Xr) is positive or negative; 
if the curve is to the left, the correction will be positive or nega- 



§ 92. EARTHWORK. 107 

tive according as (Xr—xi) is positive or negative. Therefore 
when computing curves to the right use the form (xi—Xr) in 
Eqs. 66 and 68; when computing curves to the left use the form 
(xr—xi) in these equations; the algebraic sign of the correction 
will then be strictly in accordance with the results thus obtained. 
92. Center of gravity of side-hill sections. In computing the 
correction for side-hill work the cross-section would be treated 
as triangular unless the error involved would evidently be too 
great to be disregarded. The center of gravity of the triangle 
lies on the line joining the vertex with the middle of the base 
and at J of the length of this line from the base. It is therefore 
* equal to the distance from the center to the foot of this line plus 
J of its horizontal projection. Therefore 

""4 2 "^ 3 12 ^ 6 

__6 XI Xr 

""6^3 "3 

= i[| + (..-x.)] (67) 

By the same process as that used in § 91 the correction equation 
may be wTitten 

Corr.incub.yds. = ^[F'(|-4(.rj'-a-/^) + V'^(-| + (.Tz''-a:/0)]. (68) 

It should be noted that since the grade triangle is not used in 
this computation the volume of the grade prism is not involved 
in computing the quantities F' and V'\ 

The eccentricities of cross-sections in side-hill work are never 
zero, and are frequently quite large. The total volume is gen- 
erally quite small. It follows that the correction for curvature 
is generally a vastly larger proportion of the total volume than 
in ordinary three-level or irregular sections. 

If the triangle is wholly to one side of the center, Eq. 67 can 
still be used. For example, to compute the eccentricity of the 
triangle of fill, Fig. 57, denote the two distances to the slope- 



108 RAILROAD CONSTRUCTION. § 93. 

stakes by yr and —yi (note the minus sign). Applying Eq. 67 
literally (noting that — must here be considered as negative in 
order to make the notation consistent) we obtain 



1 






which reduces to 






(69) 



As the algebraic signs tend to create confusion in these 
formulae, it is more simple to remember that for a triangle 
lying on both sides of the center e is always numerically equal 

tO"i ^-\-{xi'^Xj) L and for a triangle entirely on one side, e is 

1 rh 
numerically equal to— ^+^h^^^^^™^^ sumoi the two dis- 
tances out]. The algebraic sign of e is readily determinable as 
in § 91. 

93. Example of curvature correction. Assume that the fill in 

§ 79 occurred on a 6° curve to the right. -^ = _^-, . The 

Stt 2855 

quantities 210, 507, etc., represent the quantities V , F", etc., 

since they include in each case the 61 cubic yards due to the 

grade prism. Then 



Vjxir^xr) _ 210(22.9-8.2) ^ 3101.7 __ 
3R 2865 2865 ~ "^ 



The sign is plus, since the center of gravity of the cross-sec- 
tion is on the left side of the center and the road curves to the 
right, thus making the true volume larger. For Sta. 18 the 
correction, computed similarly, is +3, and the correction for 
the whole section is 1+3=4. For Sta. 18 + 40 the correction 
is computed as 6 yards. Therefore, for the 40 feet, the correc- 
tion is tVo(3 + 6) =3.6, which is called 4. Computing the others 
similarly we obtain a total correction of + 16 cubic yards. 



§ 94. EARTHWORK. 109 

94. Accuracy of earthwork computations. The preceding 
methods give the precise volume (except where approximations 
are distinctly admitted) of the prismoids which are supposed to 
represent the volume of the earthA\'ork. To appreciate the 
accuracy necessary in cross-sectioning to obtain a given accuracy 
in volume, consider that a fifteen-foot length of the cross-section, 
which is assumed to be straight, really sags 0.1 foot, so that the 
cross-section is in error by a triangle 15 feet wide and 0.1 foot 
high. This sag 0.1 foot high would hardly be detected by the 
eye, but in a length of 100 feet in each direction it would make 
an error of volume of 1.4 cubic yards in each of the two pris- 
moids, assuming that the sections at the other ends were perfect. 
If the cross-sections at both ends of a prismoid were in error by 
this same amount, the volume of that prismoid would be in error 
by 2.8 cubic yards if the errors of area were both plus or both 
minus. If one were plus and one minus, the errors would 
neutralize each other, and it is the compensating character of 
these errors which permits any confidence in the results as 
obtained by the usual methods of cross-sectioning. It demon- 
strates the utter futility of attempting any closer accuracy than 
the nearest cubic yard. It will thus be seen that if an error 
really exists at any cross-section it involves the prismoids on 
both sides of the section, even though all the other cross-sections 
are perfect. As a further illustration, suppose that cross-sec- 
tions were taken by the method of slope angle and center depth 
(§ 74), and that a cross-section, assumed as uniform, sags 0.4 
foot in a width of 20 feet. Assume an equal error (of same 
sign) at the other end of a 100-foot section. The error of 
volume for that one prismoid is 38 cubic 3^ards. 
f The computations further assume that the warped surface, 
passing through the end sections, coincides with the surface of 
the ground. Suppose that the cross-sectioning had been done 
with mathematical perfection; and, to assume a simple case, 
suppose a sag of 0.5 foot between the sections, which causes an 
error equal to the volume of a pyramid having a base of 20 feet 
(in each cross-section) times 100 feet (between the cross-sec- 
tions) and a height of 0.5 foot. The volume of this pyramid is 
J(20X100)X0.5 = 333 cub. ft. = 12 cub. yds. And yet this sag 
or hump of 6 inches would generally be utterly unnoticed, or 
at least disregarded. 

When the ground is very rough and broken it is sometimes 



110 



RAILROAD CONSTRUCTION. 



§95. 



practically impossible, even with frequent cross-sections, to 
locate warped surfaces which will closely coincide with all the 
sudden irregularities of the ground. In such cases the compu- 
tations are necessarily more or less approximate and dependence 
must be placed on the compensating character of the errors. 

95. Approximate computations from profiles. When a 
''paper location" has been laid out on a topographical map 
having contours, it is possible to compute approximately the 
amount of earthwork required by some very simple and rapid 
calculations. A profile may be readily drawn by noting the 
intersections of the proposed center line with the various con- 
tours and plotting the surface line on profile paper. Drawing 
the grade-line on the profile, the depth of cut or fill may be 
scaled off at any point. When it is only desired to obtain 




Fio. 58. 



very quickly an approximate estimate of the amount of earth- 
work required on a suggested line, it may be done by the method 
described in § 75. or by the use of Table XXXIII. But the 
assumption that the surface of the ground at each cross-section 
is level invariably has the effect that the estimated volumes 
are not as large as those actually required. The difference 
between the ''level section'' hkms and the actual slope section 
hknq equals the difference between the triangles mon and ogs, 
and this difference equals the shaded area mpn. The excess 
volume is proportional to the area of the triangle mpn. This 
area may be expressed by the formula, 



Area mpn = 2{hb + d cot /?) 



jSin^a sin/? cos/? 
cos 2a — cos 2/?' 



§ 95. EARTHWORK. Ill 

The percentage of this excess area to the nominal area hkms 
therefore depends on the dimensions b and d and the angles a 
and ^. A solution of this equation for ninety different com- 
binations of various numerical values for these four variables 
is included in Table XXXIII for the purpose of making cor- 
rections. A study of this correction table points conclusively 
to the following laws, a thorough understanding of which will 
enable an engineer to appreciate the degree of accuracy which 
is attainable by this approximate method: 

(a) Increasing the ividth of the roadbed (b), the other three 
factors remaining constant, increases the percentage of error, 
but the increase is comparatively small. 

(b) Increasing the depth of cut or fill (d)j decreases the per- 
centage of error, but the decrease is almost insignificant. 

(c) Increasing the angle of the side slopes (/9) decreases the 
percentage of error, the decrease being very considerable. 

(d) Increasing the angle of the slope of the ground (a), 
increases the percentage of error, the percentage rapidly in- 
creasing to infinity as the value of a approaches that of /?. 
This is another method of stating the fact that a must always 
be less than /? and, practically, must be considerably less, so 
that the slope stake shall be within a reasonable distance from 
the center. 

Since the above value for the corrective area is a function of 
the angle a, which is usually variable and whose value is fre- 
quently know^n only approximately, it is useless to attempt 
to apply the correction with great precisioiji, and the following 
rules will usually be found amply accurate, considering the 
probable lack of precision in the data used. 

1. For embankments or cuts, having a slope of 1.5:1, and 
with a surface slope of 5° (nearly 9%) the excess of true area 
over nominal area is about 2%. There is only a slight varia- 
tion from this value for all ordinary depths (d) and widths (b) 
of roadbed. Therefore the nominal volume would be about 2% 
too small. On the other hand, the effect of the prismoidal 
correction is such that, even with truly level sections, the 
nominal volume is too large. See §§75 and 76. The amount 
of the prismoidal correction depends on the differences between 
successive center depths. In the very ordinary numerical 
case given in § 76, the correction was nearly 3%, which more 
than neutralizes the error due to surface slope. Therefore in 



112 RAILROAD CONSTRUCTION. §95. 

many cases on slightly sloping ground the error due to the 
surface slope will so nearly neutralize the prismoidal correc- 
tion that the quantities taken directly from the tables (without 
correction for either cause) will equal the true volume with as 
close an approach to accuracy as the precision of the surveying 
will permit. 

2. For a cut with a slope of 1:1, and with a surface slope of 
5° the error is about 1%. This will be neutralized by still 
smaller prismoidal corrections. Therefore, for surface slopes 
of 5° or less, no allowance should be made for this error unless 
the prismoidal correction is also considered. 

3. When the surface slope is 10° (nearly 18%) the error for 
a 1.5:1 slope is from 7% to 10% and for a 1:1 slope from 3% 
to 5%. 

4. For a 30° surface slope and 1.5:1 side slopes the excess 
volume is three or four times the nominal volume. Such a 
steep surface slope implies the probability of *' side-hill work" 
to which the above corrective rules are not applicable. When 
the surface slopes are very steep careful work must be [done 
to avoid excessive error. For a 1 : 1 side slope, the errors are 
from 50% to 80%. 

A still closer approximation, especially for the steeper surface 
slopes, may be obtained by using, directly or by interpolation, 
figures from the corrective tabular form which forms part of 
Table XXXIII. Unless the surface slope angle is known 
accurately (especially when large) no great accuracy in the 
final result is possible. Close accuracy would also require the 
determination of the prismoidal correction. But if such close 
accuracy is deemed essential, it can be most easily obtained 
by accurate cross-sectioning at each station and the adoption 
of other methods of computation — such as are given in §§83 
and 84. 

When the contours have been drawn in for a sufficient 
distance on either side to include the position of both slope 
stakes at every station, as will usually be the case, cross-sections 
may be obtained by drawing lines on the map at each station 
perpendicular to the center line — see Fig. 4. The intersection 
of these lines with the contours will furnish the distances for 
drawing on cross-section paper the transverse profile at each 
station. Drawing on the same cross-section the lines repre- 
senting the roadbed and the side slopes, the cross-section of 



§95. EARTHWORK. 113 

cut (or fill) is complete and its area may be obtained by scaling 
from the cross-section paper. If the contours have been 
located on the map with sufficient accuracy, such a method 
will determine the cross-sectional area very closely. When 
cross-sections have been taken with a wye- or hand-level, as 
described in § 12, the cross-sections as plotted will probably 
be more accurate than when the contours are run in from 
points determined by the stadia method. In fact this semi- 
graphical method is frequently used, in place of the purely 
numerical methods described in previous sections, to make 
final estimates of the volume of earthwork. 

As a numerical example, an assumed location line was laid 
out on the contours given in Fig. 4. The volume of cut, as 
determined by Table XXXIII for a roadbed 20 feet wide, with 
side slopes of 1:1, was 5746 cubic yards. The surface slope 
varied from 3° to 11°. Computing the corrections by a careful 
interpolation from the corrective table, the total correction was 
found to be 128 mihic yards, or an average of a little over 2%. 
On the other hand the negative prismoidal correction amounts 
to 72 cubic yards, which leaves a net correction of 56 cubic 
yards — about 1%. It so happens that in this case a correction 
for curvature would tend further to wipe out this correction. 
These figures merely verify numerically the general conclusions 
stated above, although it should not be forgotten that in indi- 
vidual cases the figures taken from Table XXXIII require 
ample correction. 

The following approximate rule, for which the author is 
indebted to Mr. W. H. Edinger, is exceedingly useful when it 
is desired to rapidly determine the approximate volume of 
earthwork between two points along the road. Its great merit 
lies in the fact that it only means the memorizing of a com- 
paratively simple rule which will make it possible to make 
such computations in the field, without the use of tables. The 
rule is based on the fact that the area of any level section equals 
bd + sd^ ; and therefore, 

S(vol.) = (6S(^ + sSd2)^, 

in which 7^ is usually 100 feet. For strict accuracy this would 
only be the volume provided the total length was an even num- 
ber of hundred feet, and the various values of d represented 



114 RAILBOAD CONSTRUCTION. § 96 

the depths which were uniform for hundred foot sections. It 
makes no allowance for the comparatively large prismoidal 
error of the pyramidal and wedge-shaped sections usually found 
at each end of a cut or fill, but where an approximate estimate 
is desired, in which this inaccuracy may be neglected, the 
method is very useful. The method of applying this rule with- 
out tables may best be illustrated by a simple numerical ex- 
ample. Assume that the levels on a stretch of fairly level 
ground, which is about 500 feet long, have been taken, the depths 
being taken at points 100 feet apart, the first and last points 
being about 40 or 50 feet from the ends of the cut, or fill. The 
depths are as given in the first column in the tabular form 
below; the slope is 1.5:1, and the breadth (b) is 14 feet. 



d 




d' 


1.6 




2.56 


2.8 




7.84 


4.5 




20.25 


3.1 




9.61 


0.9 


HcP- 


.81 


7:d= 12.9 


= 41.07 


14 


sXd^ = 


20.53 


fc2^/=180.6 


= 61.60 


61.60 







242.2 

24220 ^ 27 = 897 cubic yards. 

The 180.6 is the b^d and the 61.6 is sSc?^; adding these and 
moving the decimal point two places to multiply by 100, we 
only have to divide by 27 to obtain the value in cubic yards. 
Although the above rule requires more work than the employ- 
ment of earthwork tables, yet it is a very convenient method 
of estimating the approximate volume of a short section of 
earthwork when no tables are at hand. 

96. Shrinkage of earthwork. The statistical data indicating 
the amount of shrinkage is very conflicting, a fact which is 
probably due to the following causes: 

1. The various kinds of earthy material act very differently 
as respects shrinkage. There is a great lack of uniformity in 



§ 96. EARTHWORK. 115 

the classification of earths in the tests and experiments which 
have been made. 

2. Very much depends on the method of forming an embank- 
ment (as will be shown later). Different reports have been 
based on different methods — often without mention of the 
method. 

3. An embankment requires considerable time to shrink to 
its final volume, and therefore much depends on. the time 
elapsed between construction and the measurement of what is 
supposed to be the settled volume. 

4. A soft subsoil will frequently settle under the weight of a 
high embankment and apparently indicate a far greater shrink- 
age than the actual reduction in volume. 

5. iVn embankment of very soft material will sometimes 
*^mush" or widen at the sides, with a consequent settling of 
the top, due to this cause alone. 

This subject has called forth much discussion in the technical 
press and literature. Quotations can be made of figures cover- 
ing a large range' of values, but space will only permit the 
statement of the conclusions which may be drawn from the 
large mass of testimony which has been presented. 

1. Volume of loose m^aterial. When material of any character 
is excavated and deposited loosely in a pile, its volume is 
always largely in excess of the volume of the excavation. 
Solid rock will occupy from 60% to 80% more space when 
broken up than when solid. A soft earth will have an excess 
volume of about 20% to 25%. 

2. Effect of method of depositing. When material is de- 
posited loosely, as from a trestle, the excess of volume when 
the embankment is just completed is very large. The time 
required for final settlement is also very great. When an 
embankment is formed by the wheelbarrow method, the initial 
expansion is about as great as when the material is merely 
dumped from ears. When the material is deposited in small 
increments from wagons and each layer is subjected to com- 
pression from horses' hoofs and from wheels, the contraction 
during construction is far greater and the additional shrinkage 
is comparatively small. Wheeled scrapers and drag scrapers 
will produce even more initial compression. 

3. Time required for final settlement. This depends partly 
on the method of formation and also on the character of the 



116 RAILROAD CONSTRUCTION. § 97. 

material. When a soft loamy soil is deposited loosely, the dry- 
ing out of the soil during the first long dry season will develop 
large cracks. Subsequent rains will close these cracks by a 
general contraction of the whole mass. When the embank- 
ment is loosely formed it may take two years before additional 
settlement becomes inappreciable, but w^hen the method of 
deposition ensures compression during construction the subse- 
quent shrinkage is less in time as well as amount. 

4. Classification of soils with respect to shrinkage. Loose 
vegetable surface soil will expand very greatly when excavated 
and first deposited, but will subsequently shrink to considerably 
less than its original volume. Clay soils are next in order 
and the sandy and gravelly soils come at the other end of 
the list of earthy materials. Rock expands very greatly when 
first broken up and deposited and there is no appreciable sub- 
sequent shrinkage. 

97. Proper allowance for shrinkage. Specifications for the 
Mississippi River levees require that there shall be a 10% 
shrinkage allowance for embankments formed by team work 
and 25% allowance for wheelbarrow work. It is contended 



r I 

A N 

'^miiiimiiniiiimiiiiiiiiiifh 



Fia. 59. 

that such figures are only justified because the subsoil settles 
or because the embankments mush out at the sides, and that 
if these effects do not occur the levees are permanently higher 
than designed. 

It is usual to require that embankments shall be constructed 
higher than their desired ultimate, as shown in Fig. 59. Since 
the base does not contract, the contraction may be said to 
be all vertical. Since a high embankment will unquestionably 
shrink a greater total amount than a low embankment (what- 
ever the percentage), it follows that an embankment having 



§ 97. EARTHWORK. 117 

variable heights (as usual) should have an initial grade-line 
somewhat like the dotted line adc in Fig. 60. Although some 
such method is essential if there is to be no ultimate sag below 
the desired grade-line, the policy is sharply criticized. The 
grade ad, even though temporary, may prove objectionable 
from an operating standpoint. Frequently the allowance is 
made too great or the shrinkage is not as much as anticipated, 
and it becomes necessary to cut off the top of the bank. On 
the other hand, the expense of raising the track after the road 
is in operation and the inevitable loss of ballast is so great 
that the danger of being required to fill u]) a sag should be 
avoided if possible. 



Fig. 60. 

A sharp and clear distinction should be made between the 
coefficient of extra height of an embankment and the coefficient 
of shrinkage which determines how many cubic yards of settled 
embankment may be made from a definite volume of earth or 
rock measured in the excavation. The values quoted above 
for the Mississippi levees (from 10% to 25%) refers usually 
to a very soft soil and includes the effects other than actual 
contraction of volume. From 8% to 15% is usually quoted as 
the required extra height of embankments, although it is 
strenuously claimed by many that 3% or 2% is sufficient, 
or even that no allowance should be made. 

The coefficients to determine the amount of settled embank- 
ment which may be made from a given volume of earth or 
rock measured in the excavation, are necessarily subject to 
variation on account of the method employed and the amount 
of compression and settlement which will take place during 
the progress of the work. The following figures have the 
weight of considerable authority but, if in error, the coefficients 
are probably high rather than low: 



118 



RAILROAD CONSTRUCTION. 



§98. 



Gravel or sand about 8% 

Clay '' 10% 

Loam '' 12% 

Loose vegetable surface soil *' 15% 

It may be noticed from the above table that the harder and 
cleaner the material the less is the contraction. Perfectly clean 
gravel or sand would not probably change volume appreciably. 
The above coefficients of shrinkage and expansion may be used 
to form the following convenient table: 



Material. 


To make 1000 cubic 

yards of embankment 

will reciuire 


1000 cubic yards 
measured in exca- 
vation will make 


Gravel or sand 


1087 cubic yards 

nil " 

11.36 •• 

1176 ** 

714 " 

625 *' 

measured in excavation 


920 cubic yards 


Clay 


900 " 


Loam 


880 * * 


Loose vegetable soil 

Rock, large pieces 

small " 


850 •• 
1400 " 
1600 " 
of embankment. 



Since writing the above the following values have been 
adopted by the American Railway Engineering and Mainte- 
nance of Way Association as representing standard practice: 

Coefficients of Shrinkage Allowance for Depositing 
Earthwork. 



Trestle filling. 


Raising under traffic. 


Black dirt 


15% 

10% 

6% 


5% 


Clay 


5% 


Sand 


5% 







98. Methods of forming embankments. Embankments of 
moderate height are sometimes formed by scraping material 
with drag scrapers from ditches at the sides, especially if there 
is little or no cutting to be done in the immediate vicinity. 
Over a low level swampy stretch this method has the double 
advantage of building an embankment which is well above 
the general level and also provides generous drainage ditches 
which keep the embankment dry. Wheeled scrapers may be 
used economically up to a distance of 400 feet to excavate 



§ 98. EARTHWORK. 119 

cuts and deposit the material on low embankments. Such 
methods have the advantage of compacting the embankments 
during construction and reducing future shrinkage. 

When carts are used, an embankment of any height may be 
formed by ''dumping over the end" and building to the full 
height (or even higher to allow for shrinkage) as the embank- 
ment proceeds. The method is especially applicable when the 
material comes from a place as high as or higher than the 
grade-line, so that no up-hill hauling is necessary. Only a 
small contractor's plant is required for all of these methods. 

Trestles capable of carrying carts, or even cars and loco- 
motives, from which excavated material may be dropped, are 
found to be economical in spite of the fact that their cost is a 
construction expense. There is the disadvantage that such 
embankments require a long time to settle, but there are the 
advantages that the earth may be hauled by the train load 
from a distance of perhaps several miles, dumped from the 




Fig. 61. 

cars by train ploughs, or automatically dumped when the 
material is carried in patent dumping-cars, and all at a com- 
paratively small cost per cubic yard. The disadvantages of 
slow settlement may be obviated, although at some additional 
cost, by making the trestle sufficiently strong to support regular 
traffic until the settlement is complete. 

During recent years cableways have been utilized to fill 
comparatively narrow but deep ravines from material obtain- 
able on either side of the ravine. This method obviates the 
construction of an excessively high trestle which might other- 
wise be considered necessary. 

When an embankment is to be placed on a steep side hill 
which has a slippery clay surface, the embankment will some- 



120 RAILROAD CONSTRUCTION. § 98. - 

times slide down the hill, unless means are taken to prevent it. 
Some sort of bond between the old surface and the new material 
becomes necessary. This has sometimes been provided by 
cutting out steps somewhat as is illustrated in Fig. 61. It is 
possible that a deep ploughing of the surface would accom- 
plish the rtsult just as effectively and much cheaper. The 
tendency to slip is generally due not only to the nature of the 
soil but also to the usual accompanying characteristic that the 
soil is wet and springy. The sub-surface drainage of such a 
place with tile drains will still further prevent such slipping, 
which often proves very troublesome and costly. 

COMPUTATION OF HAUL. 

99. Nature of subject. As will be shown later when analyz- 
ing the cost of earthwork, the most variable item of cost is that 
depending on the distance hauled. As it is manifestly imprac- 
ticable to calculate the exact distance to which every individual 
cartload of earth has been moved, it becomes necessary to devise 
a means which will give at least an equivalent of the haulage of 
all the earth moved. Evidently the average haul for any mass 
of earth moved is equal to the distance from the center of grav- 
ity of the excavation to the center of gravity of the embank- 
ment formed by the excavated material. As a rough approxi- 
mation the center of gravity of a cut (or fill) may sometimes be 
considered to coincide with the center of gravity of that part of 
the profile representing it, but the error is frequently very large. 
The center of gravity may be determined by various methods, 
but the method of the ^'mass diagram" accomplishes the same 
ultimate purpose (the determination of the haul) with all-suffi- 
cient accuracy and also furnishes other valuable information. 

100. Mass diagram. In Fig. 62 let A'B' ... 6" represent 
a profile and grade line drawn to the usual scales. Assume A' 
to be a point past which no earthwork will be hauled. Such 
a point is determined by natural conditions, as, for example, a 
river crossing, or one end of a long level stretch along which 
no grading is to be done except the formation of a low embank- 
ment from the material excavated from ample drainage ditches 
on each side. Above the profile draw an indefinite horizontal line 
{ACn in Fig. 62), which may be called the ''zero line." Above 
every station point in the profile draw an ordinate (above or be- 



§ 100. 



EARTHWORK. 



121 



hrj \?^ 



I 



low the zero line) which will represent the algebraic sum of 
the cubic yards of cut and fill 
(calling cut -f and fill — ) from 
the point A^ to the point con- 
sidered. The computations of 
these ordinates should first be 
made in tabular form as shown 
below. In doing this shrinkage 
must be allowed for by consider- 
ing how much embankment 
would actually be made by so 
many cubic yards of excavation 
of such material. For example, 
it will be found that 1000 cubic 
yards of sand or gravel, measured 
in place (see § 97), will make 
about 920 cubic yards of embank- 
ment; therefore all cuttings in 
sand or gravel should be dis- 
counted in about this propor- 
tion. Excavations in rock should 
be increased in the proper 
ratio. In short, all excavations 
should be valued according to the 
amount of settled embankment 
that could be made from them. 
Place in the first column a list 
of the stations; in the second 
column, the number of cubicyards 
of cut or fill between each station 
and the preceding station; in 

the third and fourth columns, the kind of material and the proper 
shrinkage factor; in the fifth column, a repetition of the quan- 
tities in cubic yards, except that the excavations are diminished 
(or increased, in the case of rock) to the number of cubic yards 
of settled embankment which may be made from them. In 
the sixth column place the algebraic sum of the quantities in the 
fifth column (calling cuts + and fills — ) from the starting- 
point to the station considered. These algebraic sums at each 
station will be the ordinates, drawn to some scale, of the mass 
curve. The scale to be used will depend somewhat on whether 




122 



RAILROAD CONSTRUCTION. 



§ 101. 



the work is heavy or hght, but for ordinary cases a scale of 
5000 cubic yards per inch may be used. Drawing these ordi- 
nates to scale, a curve A, B, . . . G may be obtained by joining 
the extremities of the ordinates. 



Sta. 


Yards] -tl 


Material. 


Shrinkage 
factor. 


Yards, 

reduced 

for 

shrinkage. 


Ordinate 
in mass 
curve. 


46 4- 70 













47 
48 

4- 60 
49 


4- 195 
4- 1792 
4- 614 

- 143 

- 906 

- 1985 

- 1721 

- 112 
+ 177 
4- 180 

- 52 

4- 276 
4- 1242 
4- 1302 


Clayey soil 


- 10 per cent 

- 10 

- 10 


4- 175 
4- 1613 
4- 553 

- 143 

- 906 

- 1985 

- 1721 

- 112 
+ 283 
4- 289 

- 52 

- 71 
4- 249 
4- 1118 
4- 1172 


4- 175 

4- 1788 
4- 2341 
4- 2198 


50 






4- 1292 


51 






- 693 


52 






- 2414 


4- 30 






- 2526 


53 

4- 70 
54 


Hard rock 


+ 60 per cent 
4- 60 


- 2243 

- 1954 

- 2006 


4- 42 







- 2077 


55 
56 
57 


Clayey soil 


- 10 per cent 

- 10 

- 10 


- 1828 

- 710 
4- 462 



loi. Properties of the mass curve. 

1. The curve will be rising while over cuts and falling while 
over fills. 

2. A tangent to the curve will be horizontal (as at B, D, E, 
F, and G) when passing from cut to fill or from fill to cut. 

3. When the curve is below the "zero line" it shows that 
material must be drawn backward (to the left) ; and vice versa, 
when the curve is above the zero line it shows that material 
must be drawn forward (to the right) . 

4. When the curve crosses the zero line (as at A and C) it 
shows (in this instance) that the cut between A' and B' will just 
provide the material required for the fill between B' and C, and 
that no material should be hauled past C, or, in general, past 
any intersection of the mass curve and the zero line. 

5. If any horizontal line be drawn (as ab), it indicates that 
the cut and fill between a' and b' will just balance. 

6. When the center of gravity of a given volume of material 
is to be moved a given distance, it makes no difference (at least 
theoretically) how far each individual load may be hauled or 
how any individual load may be disposed of. The summation 



§ 101. EARTHWORK. 123 

of the products of each load times the distance hauled will be a 
constant, whatever the method, and will equal the total volume 
times the movement of the center of gravity. The average 
haul J which is the movement of the center of gravity, will there- 
fore equal the summation of these products divided by the total 
volume. If we draw two horizontal parallel lines at an infini- 
tesimal distance dx apart, as at ah, the small increment of cut 
dx at a' will fill the corresponding increment of fill at b\ and 
this material must be hauled the distance ah. Therefore the 
product of ah and dx, which is the product of distance times 
volume, is represented by the area of the infinitesimal rectangle 
at oh, and the total area ABC represents the summation of 
volume times distance for all the earth movement between A' 
and C\ This summation of products divided by the total 
volume gives the average haul. 

7. The horizontal line, tangent at E and cutting the curve 
at e, f, and g, shows that the cut and fill betw^een e' and E^ will 
just balance, and that a possible method of hauling (whether 
desirable or not) would be to " borrow" earth for the fill between 
C and e', use the material between Z)' and £" for the fill between 
e' and D', and similarly balance cut and fill between E' and /' 
and also between /' and g\ 

8. Similarly the horizontal line hklm may be dra^n cutting 
the curve, which will show another possible method of hauling. 
According to this plan, the fill between C and N would be 
made by borrowing; the cut and fill between h' and k' would 
balance; also that between k' and V and between V and m'. 
Since the area ehDkE represents the measure of haul for the 
earth between e' and £", and the other areas measure the corre- 
sponding hauls similarly, it is evident that the sum of the areas 
ehDkE and ElFmf, w^hich is the measure of haul of all the 
material between e' and /', is largely in excess of the sum of 
the areas hDk, kEl, and IFm, plus the somewhat uncertain 
measures of haul due to borrowing material for e'/i' and wasting 
the material between m' and /'. Therefore to make the meas- 
ure of haul a minimum a line should be drawn which will make 
the sum of the areas between it and the mass curve a minimum. 
Of course this is not necessarily the cheapest plan, as it implies 
more or less borrowing and wasting of material, which may 
cost more than the amount saved in haul. The comparison of 
the two methods is quite simple, however. Since the amount 



124 RAILROAD CONSTRUCTION. § 102. 

of fill between e' and h' is represented by th.e difference of the 
ordinates at e and h, and similarly for m' and /', it follows that 
the amount to be borrowed between e' and h' will exactly equal 
the amount wasted between m' and /'. By the first of the above 
methods the haul is excessive, but is definitely known from the 
mass diagram, and all of the material is utilized; by the second 
method the haul is reduced to about one-half, but there is a 
known quantity in cubic 3^ards wasted at one place and the same 
quantity borrowed at another. The length of haul necessary 
for the borrowed material would need to be ascertained; also 
the haul necessary to waste the other material at a place where 
it would be unobjectionable. Frequently this is best done by 
widening an embankment beyond its necessary width. The 
computation of the relative cost of the above methods will be 
discussed later (§ 116). 

9. Suppose that it were deemed best, after drawing the mass 
curve, to introduce a trestle between s' and v' , thus saving an 
amount in fill equal to tv. If such had been the original design, 
the mass curve would have been a straight horizontal line between 
s and t and would continue as a curve which would be at all 
points a distance tv above the curve vFmzfGg. If the line Ef is 
to be used as a zero line, its intersection with the new curve at x 
will show that the material between E^ and 2' will just balance 
if the trestle is used, and that the amount of haul will be meas- 
ured by the area between the line Ex and the broken line Estx. 
The same computed result may be obtained without drawing 
the auxiliary curve txn ... by drawing the horizontal line zy 
at a distance xz{=tv) below Ex. The amount of the haul can 
then be obtained by adding the triangular area between Es and 
the horizontal line Ex, the rectangle between st and Ex, and the 
irregular area between vFz and y . . . z (which last is evidently 
equal to the area between tx and E . . . x). The disposal of the 
material at the right of 2' would then be governed by the indica- 
tions of the profile and mass diagram which would be found at 
the right of g\ In fact it is difficult to decide with the best of 
judgment as to the proper disposal of material without having 
a mass diagram extending to a considerable distance each side 
of that part of the road under immediate consideration. 

102. Area of the mass curve. The area may be computed 
most readily by means of a planimeter, which is capable with 
reasonable care of measuring such areas with as great accuracy 



§ 103. EARTHWORK. 125 

as is necessary for this work. If no such instrument is obtain- 
able, the area may be obtained by an application of ''Simpson's 
rule." The ordinates will usually be spaced 100 feet apart. 
Select an even number of such spaces, leaving, if necessary, one 
or more triangles or trapezoids at the ends for separate and 
independent computation. Let i/o • • • ^n be the ordinates, i.e., 
the number of cubic yards at each station of the mass curve, or 
the figures of "column six" referred to in § 100. Let the uni- 
form distance between ordinates (^100 feet) be called 1, i.e., 
one station. Then the units of the resulting area will be cubic 
yards hauled one station. Then the 

Area = i[?/o+ 4(2/1 + 2/3 -^...2/(w-l) + 2(2/2 + 2/4 +...2/(n_2) + 2/n]- (70) 

When an ordinate occurs at a substation, the best plan is to 
ignore it at first and calculate the area as above. Then, if the 
difference involved is too great to be neglected, calculate the 
area of the triangle having the extremity of the ordinate at the 
substation as an apex, and the extremities of the ordinates at the 
adjacent stations as the ends of the base. This may be done by 
finding the ordinate at the substation that would be a propor- 
tional between the ordinates at the adjacent full stations. Sub- 
tract this from the real ordinate (or vice versa) and multiply the 
difference by J XL An inspection will often show^ that the 
correction thus obtained would be too small to be worthy of con- 
sideration. If there is more than one substation between two 
full stations, the corrective area will consist of two triangles and 
one or more trapezoids which may be similarly computed, if 
necessary. 

When the zero line (Fig. 62) is shifted to eE, the drop from- 
AC (produced) to E is known in the same units, cubic yards. 
This constant may be subtracted from the numbers ("column 
6," § 100) representing the ordinates, and will thus give, with- 
out any scaling from the diagram, the exact value of the modi- 
fied ordinates. 

103. Value of the mass diagram. The great value of the mass 
diagram lies in the readiness with which different plans for the 
disposal of material may be examined and compared. When 
the mass curve is once drawn, it will generally require only a 
shifting of the horizontal line to show the disposal of the material 
by any proposed method. The mass diagram also shows the 



126 RAILROAD CONSTRUCTION. § 104. 

extreme length of haul that will be required by any proposed 
method of disposal of material. This brings into consideration 
the ''limit of profitable haul," which wnll be fully discussed in 
§ 116. For the present it may be said that with each method 
of carrying material there is some limit beyond which the expense 
of hauling wull exceed the loss resulting from borrowing and 
wasting. With wheelbarrows and scrapers the limit of profit- 
able haul is comparatively short, with carts and tram-cars it is 
much longer, w^hile w^ith locomotives and cars it may be several' 
miles. If, in Fig. 62, eE or Ef exceeds the limit of profitable 
haul, it shows at once that some such line as hklm should be 
drawn and the material disposed of accordingly. 

104. Changing the grade line. The formation of the mass 
curve and the resulting plans as to the disposal of material are 
based on the mutual relations of the grade line and the surface 
profile and the amounts of cut and fill which are thereby im- 
plied. If the grade line is altered, every cross-section is altered, 
the amount of cut and fill is altered, and the mass curve is also 
changed. At the farther limit of the actual change of the grade 
line the revised mass curve will have (in general) a different 
ordinate from the previous ordinate at that point. From that 
point on, the revised mass curve will be parallel to its former 
position, and the revised curve may be treated similarly to the 
ease previously mentioned in which a trestle w^as introduced. 
Since it involves tedious calculations to determine accurately 
how much the volume of earthwork is altered by a change in 
grade line, especially through irregular country, the effect on 
the mass curve of a change in the grade line cannot therefore 
be readily determined except in an approximate way. Raising 
the grade line will evidently increase the fills and diminish the 
cuts, and vice versa. Therefore if the mass curve indicated, for 
example, either an excessively long haul or the necessity for 
borrowing material (implying a fill) and wasting material 
farther on (implying a cut), it would be possible to diminish the 
fill (and hence the amount of material to be borrowed) by lower- 
ing the grade line near that place, and diminish the cut (and 
hence the amount of material to be wasted) by raising the 
grade line at or near the place farther on. Whether the advan- 
tage thus gained would compensate for the possibly injurious 
effect of these changes on the grade line would require patient 
investigation. But the method outlined shows how the mass 



§ 105 



EARTHWORK. 



127 



curve might be used to indicate a possible change in grade line 
which might be demonstrated to be profitable. 

105. Limit of free haul. It is sometimes specified in con- 
tracts for earthwork that all material shall be entitled to free 
haul up to some specified limit, say 500 or 1000 feet, and that 
all material drawn farther than that shall be entitled to an 
allowance on the excess of distance. It is manifestly imprac- 
ticable to measure the excess for each load, as much so as to 
measure the actual haul of each load. The mass diagram also 
solves this problem very readily. Let Fig. 63 represent a pro- 




FiG. 63. 



file and mass diagram of about 2000 feet of road, and suppose 
that 800 feet is taken as the hmit of free haul. Find two points, 
a and h, in the mass curve which are on the same horizontal line 
and which are 800 feet apart. Project these points down to a' 
and h\ Then the cut and fill between a^ and y will just balance, 
and the cut between A ' and a' will be needed for the fiU between 
h' and C In the mass curve, the area between the horizontal 
line ah and the curve aBh represents the haulage of the material 
between a' and V , which is all free. The rectangle ahmn repre- 
sents the haulage of the material in the cut A' a' across the 800 
feet from a' to h\ This is also free. The sum of the two areas 
Aam and hnC represents the haulage entitled to an allowance, 
since it is the summation of the products of cubic yards times 
the excess of distance hauled. 

If the amount of cut and fill was symmetrical about the point 



128 RAILROAD CONSTRUCTION. § 106. 

B'y the mass curve would be a S3^mmetrical curve about the 
vertical line through B, and the two limiting lines of free haul 
would be placed symmetrically about B and B' . In general 
there is no such symmetry, and frequently the difference is con- 
siderable The area ciBbnm will be materially changed accord- 
ing as the two vertical lines am and hn, always 800 feet apart, 
are shifted to the right or left. It is easy to show that the area 
aBbnm is a maximum when ab is horizontal. The minimum 
value would be obtained either when m reached A or n reached 
Cy depending on the exact form of the curve. Since the posi- 
tion for the minimum value is manifestly unfair, the best definite 
value obtainable is the maximjum, which must be obtained as 
above described. Since aBbnm is made maximum, the remainder 
of the area, which is the allowance for overhaul, becomes a mini- 
mum. The areas Aam and bCn may be obtained as in § 102. 
If the whole area AaBbCA has been previously computed, it 
may be more convenient to compute the area aBbnm and sub- 
tract it from the total area. 

Since the intersections of the mass curve and the "zero line" 
mark limits past which no material is drawn, it follows that 
there will be no allowance for overhaul except where the dis- 
tance between consecutive intersections of the zero line and mass 
curve exceeds the limit of free haul. 

Frequently all allowances for overhaul are disregarded; the 
profiles, estimates of quantities, and the required disposal of 
material are shown to bidding contractors, and they must then 
make their own allowances and bid accordingly. This method 
has the advantage of avoiding possible disputes as to the amount 
of the overhaul allowance, and is popular with railroad com- 
panies on this account. On the other hand the facility with 
which different plans for the disposal of material may be studied 
and compared by the mass-curve method facilitates the adoption 
of the most economical plan, and the elimination of uncertainty 
will frequently lead to a safe reduction of the bid, and so the 
method is valuable to both the railroad company and the con- 
tractor. 

ELEMENTS OF THE COST OF EARTHWORK. 

io6. Analysis of the total cost into items. The variation in 
the total cost of excavating earthwork, hauling it a greater or 
less distance, and forming W'ith it an embankment of definite 



§ 107, EARTHWORK. 129 

• 
form or wasting it on a spoil bank, is so great that the only 
possible method of estimating the cost under certain assumed 
conditions is to separate the total cost into elementary items. 
Ellwood Morris was perhaps the first to develop such a method 
— see Journal of the Franklin Institute, September and October, 
1841. Trautwine used the same general method with some 
modifications. The following analysis will follow the same 
general plan, will quote some of the figures given by Morris 
and by Trautwine, but will also include facts and figures better 
adapted to modern conditions. Since every item of cost (except 
interest on cost of plant and its depreciation) is a direct function 
of the current price of common labor, all calculations will be 
based on the simple unit of $1 per day. Then the actual cost 
may be obtained by multiplying the calculated cost under the 
given conditions by the current price of day labor. When 
possible, figures will be quoted giving the cost of all items of 
work on a loose sandy soil which is the easiest to work and also 
for the cost of the heaviest soils, such as stiff clay and hard pan. 
These represent the extremes, excluding rock, which will be 
treated separately. The cost of intermediate grades may be 
interpolated between the extreme values according to the 
judgment of the engineer as to the character of the soil. 

The possible division into items varies greatly according to 
the method adopted, but the differentiation into items given 
below (which is strictly applicable to the old fashioned simpler 
methods of work) can usually be applied to any other method 
by merely combining or eliminating some of the items. The 
items are 

1. Loosening the natural goil. 

2. Loading the soil into whatever carrier may be used. 

3. Hauling excavated material from excavation to embank- 

ment or spoil bank. 

4. Spreading or distributing the soil on the embankment. 

5. Keeping roadways or tracks in good running order. 

6. Trimming cuts to their proper cross-section (sometimes 

called '^sandpapering")- 

7. Repairs, wear, depreciation, and interest on cost of plant. 

8. Superintendence and incidentals. 

107. Item I. Loosening, (a) Ploughs. Very light sandy 
soils can frequently be shovelled without any previous loosen- 
ing, but it is generally economical, even with very light material, 



130 RAILROAD CONSTRUCTION. § 107. 

to use a plough. Morris quotes, as the results of experiments, 
that a tkree-horse plough would loosen from 250 to 800 cubic 
yards of earth per day, which at a valuation of $5 per day 
would make the cost per yard vary from 2 cents to 0.6 cent. 
Trautwine estimates the cost on the basis of two men handling 
a two-horse plough at a total cost of $3.87 per day, being SI 
each for the men, 75 c. for each horse, and an allowance of 
37 c. for the plough, harness, etc. From 200 to 600 cubic yards 
is estimated as a fair day's work, which makes a cost of 1.9 c. 
to 0.65 c. per yard, which is substantially the same estimate 
as above. Extremely heavy soils have sometimes been loosened 
by means of special ploughs operated by traction-engines. 

Gillette estimates that ''a two-horse team with a driver and 
a man holding the plough will loosen 25 cubic yards of fairly 
tough clay, or 35 cubic yards of gravel and loam per hour." 
For ten hours per day this would be 250 to 350 cubic yards 
per day. These values are neither as high nor as low as the 
extremes above noted. It is probably very seldom that a soil 
will be so light that a two-horse (or three-horse) plough cap 
loosen as much as 600 (or 800) cubic yards per day. 

It is sometimes necessary to plough up a macadamized street. 
This may be done by using as a plough a pointed steel bar 
which is fastened to a very strong plough frame. A prelimi- 
nary hole must be made which will start the bar under the 
macadam shell. Then, as the plough is drawn ahead, the shell 
is ripped up. Four or six horses, or even a traction-engine, 
are used for such work. Gillette quotes two such cases where 
the cost of such loosening was 2 c. and 6 c. per cubic yard, 
with common labor at 15 c. per hour. Two-thirds of such 
figures will reduce them to the $1 per day basis. The cost for 
ploughing on the $1 "per day basis may therefore be summarized 
as follows: 

For very loose sandy soils 0.6c. per cubic yard 

'' '' heavy clay '' 2.0 c. '' '' '' 

'* hard pan and macadam, up to .. . 4.0 c. *' " " 

(b) Picks. When picks are used for loosening the earth, as 
is frequently necessary and as is often done when ploughing 
would perhaps be really cheaper, an estimate * for a fair day's 

* Traut\Ndne. 



§ 108. EARTHWORK. 131 

work is from 14 to 60 cubic yards, the 14 yards being the esti- 
mate for stiff clay or cemented gravel, and the 60 yards the esti- 
mate for the lightest soil that would require loosening. At $1 
per day this means about 7 c. to 1.7 c. per cubic yard, which is 
about three times the cost of ploughing. Five feet of the face 
is estimated * as the least width along the face of a bank that 
should be allowed to enable each laborer to work with freedom 
and hence economically. 

(c) Blasting. Although some of the softer shaly rocks may 
be loosened with a pick for about 15 to 20 c. per yard, yet rock 
in general, frozen earth, and sometimes even compact clay are 
most economically loosened by blasting. The subject of blast- 
ing w^ill be taken up later, §§ 117-123. 

(d) Steam-shovels. The items of loosening and loading 
merge together with this method, which will therefore be treated 
in the next section. 

io8. Item 2. Loading, (a) Hand-shovelling. Much depends 
on proper management, so that the shovellers need not wait un- 
duly either for material or carts. With the best of management 
considerable time is thus lost, and yet the intervals of rest 
need not be considered as entirely lost, as it enables the men to 
work, while actually loading, at a rate which it would be physi- 
cally impossible for them to maintain for ten hours. Seven 
shovellers are sometimes allowed for each cart; otherwise there 
should be five, two on each side and one in the rear. Economy 
requires that the number of loads per cart per day should be 
made as large as possible, and it is therefore wise to employ as 
many shovellers as can work without mutual interference and 
without wasting time in waiting for material or carts. The 
figures obtainable for the cost of this item are unsatisfactory on 
account of their large disagreements. The following are quoted 
as the number of cubic yards that can be loaded into a cart by 
an average laborer in a working day of ten hours, the lower 
estimate referring to heavy soils, and the higher to light sandy 
soils: 10 to 14 cubic yards (Morris), 12 to 17 cubic yards (Has- 
koll), 18 to 22 cubic yards (Hurst), 17 to 24 cubic yards (Traut- 
wine), 16 to 48 cubic yards (Ancelin). As these estimates are 
generally claimed to be based on actual experience, the discrep- 
ancies are probably due to differences of management. If the 

* Hurst. 



132 RAILROAD CONSTRUCTION. § 108- 

average of 15 to 25 cubic yards be accepted, it means, on the 
basis of $1 per day, 6.7 c. to 4 c. per cubic yard. These esti- 
mates apply only to earth. Rockwork costs more, not only 
because it is harder to handle, but because a cubic yard of solid 
rock, measured in place, occupies about 1.8 cubic yards when 
broken up, while a cubic yard of earth will occupy about 1.2 
cubic yards. Rockwork will therefore require about 50% more 
loads' to haul a given volume, measured in place, than will the 
same nominal volume of earthwork. The above authorities give 
estimates for loading rock varying from 6.9 c. to 10 c. per cubic 
yard. The above estimates apply only to the loading of carts 
or cars with shovels or by hand (loading masses of rock). The 
cost of loading wheelbarrows and the cost of scraper work will 
be treated under the item of hauling. 

(b) Steam-shovels.* Whenever the magnitude of the work 
will warrant it there is great economy in the use of steam-shovels. 
These have a ''bucket" or ''dipper" on the end of a long beam, 
the bucket having a capacity varying from h to 2J cubic yards. 
Steam-shovels handle all kinds of material from the softest 
earth to shale rock, earthy material containing large boulders, 
tree-stumps, etc. The record of work done varies from 200 to 
1000 cubic yards in 10 hours. They perform all the work of 
loosening and loading. Their economical working requires that 
the material shall be hauled away as fast as it can be loaded, 
which usually means that cars on a track, hauled by horses or 
mules, or still better by a locomotive, shall be used. The ex- 
penses for a steam-shovel, costing about $5000, will average 
about $1000 per month. Of this the engineer may get $100; the 
fireman $50; the cranesman $90; repairs perhaps $250 to $300; 
coal, from 15 to 25 tons, cost very variable on account of expen- 
sive hauling; water, a very uncertain amount, sometimes costing 
$100 per month; about five laborers and a foreman, the laborers 
getting $1.25 per day and the foreman $2.50 per day, which will 
amount to $227.50 per month. This gang of laborers is employed 
in shifting the shovel when necessary, taking up and relaying 

* For a thorough treatment of the capabilities, cost, and management 
of steam-snovels the reader is referred to " Steam-shovels and Steam-shovel 
Work," by E. A. Hermann. D. Van Nostrand Co., New York. 

This book is now out of print. ' Earthwork and its Cost," by H. P. Gil- 
lette, to which the student is referred for a more elaborate exposition of the 
subject, has used many of Hermann's cuts. 



§ 108. EARTHWORK. 133 

tracks for the cars, shifting loaded and unloaded cars^ etc. In 
shovelling through a deep cut^ the shovel is operated so as to 
undermine the upper parts of the cut> which then fall down 
within reach of the shovel, thus increasing the amount of material 
handled for each new position of the shovel. If the material is 
too tough to fall down by its own weight, it is sometimes found 
economical to employ a gang of men to loosen it or even blast it 
rather than shift the shovel so frequently. Non-condensing 
engines of 50 horse-power use so much water that the cost of 
water-supply becomes a serious matter if water is not readily 
obtainable. The lack of water facilities will often justify the 
construction of a pipe line from some distant source and the 
installation of a steam-pump. Hence the seemingly large 
estimate of $100 per month for water-supply, although under 
favorable circumstances the cost may almost vanish. The larger 
steam-shovels will consume nearly a ton of coal per day of 10 
hours. The expense of hauling this coal from the nearest rail- 
road or canal to the location of the cut is often a very serious 
item of expense and may easily double the cost per ton. Some 
steam-shovels have been constructed to be operated by electrioity 
obtained from a plant perhaps several miles away. Such a 
method is especially advantageous when fuel and water are diffi- 
cult to obtain. 

The following general requirements and specifications were 
recommended in 1907 by the American Railway Engineering 
and Maintenance of Way Association: 

Three important cardinal points should be given careful 
attention in the selection of a steam-shovel. These are in their 
order 

(1) Care in the selection, inspection and acceptance of all 
material that enters into every part of the machine. 

(2) Design for strength. 

(3) Design for production. 

GENERAL SPECIFICATIONS. 

Weight of shovel: Seventy (70) tons. 
Capacity of dipper: Two and one-half (2^) yards. 
Steam pressure: One hundred and twenty (120) pounds. 
Clear height above rail of shovel track at which dipper should 
unload: Sixteen (16) feet. 



134 RAILROAD CONSTRUCTION. § 109. 

Depth below rail of shovel track at which dipper should aig 
Four (4) feet. 

Number of movements of dipper per minute from time of 
entering bank to entering bank: Three (3). 

Character of hoist: Cable. 

Character of swing : Cable. 

Character of housing: Permanent for all employes. 

Capacity of tank: Two thousand (2000) gallons. 

Capacity of coal-bunker: Four (4) tons. 

Spread of jack arm: Eighteen (18) feet. A special short arm 
should be provided. 

Form of steam-shovel track: "T'* rails on ties. 

Length of rails for ordinary work: Six (G) feet. 

Form of rail joint: Strap. 

Manufacturers of steam-shovels will eometimes "guarantee" 
that certain of their shovels will excavate, say 3000 cubic yards 
of earth per day of ten hours. Even if it were possible for a 
shovel to fill a car at the rate of 5 cubic yards per minute, it is 
always impracticable to maintain such a speed, since a shovel 
must always wait for the shifting of cars and for the frequent 
shifting of the shovel itself. There are also delays due to 
adjustments and minor breakdowns. The best shovel records 
are made when the cars are large — other things being equal. 
The item of interest and depreciation of the plant is very large 
in steam-shovel work. This will be discussed further later. 
The cost of loading alone will usually come to between 3 and 
4 c. per cubic yard. The cost of shifting the cars so as to 
place them successively under the shovel, haul them to the 
dumping place, dump them and haul them back, will generally 
be as much more. Gillette quotes five jobs on one railroad 
where the total cost for loading and hauling varied from 5.9 c. 
to 11.4 c. per cubic yard. But as these figures are based on 
car measurement, the cost per cubic yard in place measure- 
ment must be increased about one-fourth, or from 7.4 c. to 
14.2 c. 

109. Item 3. Hauling. The cost of hauling depends on 
the number of round trips per day that can be made by each 
vehicle employed. As the cost of each vehicle is practically the 
same whether it makes many trips or few, it becomes important 
that the number of trips should be made a maximum, and to that 
end there should be as little delay as possible in loading and 



§ 109 EARTHWORK. 135 

unloading. Therefore devices for facilitating the passage of the 
vehicles have a real money value. 

(a) Carts. The average speed of a horse hauling a two- 
wheeled cart has been found to be 200 feet per minute, a little 
slower when hauling a load and a little faster when returning 
empty. This figure has been repeatedly verified. It means an 
allowance of one minute for each 100 feet (or '' station *') of 
^4ead — the lead being the distance the earth is hauled." The 
time lost in loading, dumping, waiting to load, etc., has been 
found to average 4 minutes per load. Representing the num- 
ber of stations (100 feet) of lead by s, the number of loads 
handled in 10 hours (600 minutes) would be 600 -^(s + 4). The 
number of loads per cubic yard, measured in the bank, is differ- 
entiated by Morris into three classes, viz. : 

3 loads per cubic yard in descending hauling; 
Si '' '' " '' " level hauling; and 

4 ' ' ' ' '^ " ^' ascending hauling. 

Attempts have been made to estimate the effect of the grade 
of the roadw^ay by a theoretical consideration of its rate, and of 
the comparative strength of a horse on a level and on various 
grades. While such computations are always practicable on a 
railway (even on a temporary construction track), the traction 
on a temporary earth roadway is always very large and so very 
variable that any refinements are useless. On railroad earth- 
work the hauling is generally nearly level or it is descending — 
forming embankments on low^ ground with material from cuts in 
high ground. The only common exception occurs when an 
embankment is formed from borrow^-pits on low ground. One 
method of allowing for ascending grade is to add to the hori- 
zontal distance 14 times the difference of elevation for work 
wdth carts and 24 times the difference of elevation for work 
with w^heelbarrow^s, and use that as the lead. For example, 
using carts, if the lead is 300 feet and there is a difference of 
elevation of 20 feet, the lead would be considered equivalent to 
300 + (14X20) =580 feet on a level. 

Trautwine assumes the average load for all classes of work 
to be J cubic yard, which figure is justified by large experience. 
Using one figure for all classes of work simplifies the calculations 
and gives the number of cubic yards carried per day of 10 hours 

equal to — —. Dividing the cost of a cart per day by the 

o(s-i-4) 



136 RAILROAD CONSTRUCTION. § 109. 

number of cubic yards carried gives the cost of hauling per 
yard. In computing the cost of a cart per day, Trautwine 
refers to the practice of having one driver manage four carts, 
thus making a charge of 25 c. per day for each cart for the driver. 
Although this might be an economical method when the haul is 
very long, it is not economical for short hauls. A safer estimate 
is to allow not more than two carts per driver and in many 
cases a driver for each cart. Some contractors employ a driver 
for each cart and then require that the drivers shall assist in 
loading. The policy to be adopted is sometimes dependent on 
labor union conditions, which may demand that drivers must 
not assist in loading. The supply of labor and the amount of 
work on hand have a great influence on the methods of work 
which a contractor may adopt, for a strike will often disarrange 
all plans. 

The cost of a horse and cart must practically include a 
charge for the time of the horse on Sundays, rainy days and 
holidays. The cost of repairs of cart and harness is generally 
included in this item for simplicity, but, under a strict applica- 
tion of the analysis suggested in § 106, it should properly be 
included under Item 7, Repairs, etc. 

Since the time required for loading loose rock is greater 
than for earthwork, less loads will be hauled per day. The time 
allowance for loading, etc., is estimated by Trautwine as 6 
minutes instead of 4 as for earth. Considering the great ex- 
pansion of rock when broken up (see § 97), one cubic yard of 
solid rock, measured in place, would furnish the equivalent of 
five loads of earthwork of ^ cubic yard. Therefore, on the 
basis of five loads per cubic yard, the number of cubic yards 

handled per day per cart would be -^t — —prr. 

5(s4-6) 

Let C represent the daily cost of a horse and cart and of 

the proportional cost of the driver (according to the number of 

carts handled by one driver), then the cost per cubic yard, 

measured in the cut, for hauling may be given by the formula: 



Cost per cu. yd. of hauling earth in carts = ^^ - 

it It a u a ic «^oL- ^' << — X5(s + 6) 

rocK ggg 



(71) 



§ 109. EARTHWORK. 137 

(b) Wagons. For longer leads (i.e., from ^ to f of a mile) 
wagons drawn by two horses have been found most economical. 
The wagons have bottoms of loose thick narrow boards and are 
unloaded very easily and quickly by lifting the individual boards 
and breaking up the continuity of the bottom, thus depositing 
the load directly underneath the wagon. The capacity is about 
one cubic yard. The cost may be estimated on the same prin- 
ciple as that for carts. 

The number of wagon trips per 10 hours will depend some- 
what on the management of the shovellers. Too many shovel- 
lers per wagon is not economical, measured in yards shovelled 
per man, although it may reduce the time consumed in loading 
any one wagon. At an average figure of 20 cubic yards, 
measured in place, per shoveller per 10 hours, seven shovellers 
would load 14 cubic yards per hour or one cubic yard in 4.3 
minutes. This w^ould be the allowance for a wagon with a 
capacity of about IJ yards of loose earth. Adding time for 
unloading, waiting to load and other possible '^ lost time/' there 
is probably a total of six minutes. This figure will vary very 
considerably according to the number of shovellers per wagon, 
the capacity of the wagon, the type of wagon (whether self- 
dumping) and other details in the method of management. 
Adopting six minutes as the time used for loading, unloading, 
and other ''lost time," the formula becomes. 

Cost per cubic yard of hauling in wagons = ^ — ^, . . . (71a) 

in which C is the cost of the wagon, team and driver per day 
of 10 hours; s is the distance hauled in stations of 100 feet, 
and c is the capacity of the wagon in cubic yards, 'place meas- 
urementf which should be about three fourths of the nominal 
capacity of the w^agon for earth and about sixty per cent when 
handling rock. 

(c) Wheelbarrows. Gillette has computed from observa- 
tions that a man will trundle a wheelbarrow at the rate of 250 
feet per minute or 1.25 stations of lead per minute for the round 
trip. The time required for loading is estimated at 2^ minutes 
and for unloading, adjusting wheeling planks, short rests, etc., 
J minute, or a total of three minutes per trip for all purposes 
except hauling. Gillette allows for a load only 1/15 cubic yard, 



138 RAILROAD CONSTRUCTION. § 109. 

measured in place, or about 1/11 yard, 2,5 cubic feet, on the 
wheelbarrow. With notation as before 

Cost per cubic yard of loading and \ _ CX15(1.25g + 3) 
hauling earth in wheelbarrows / 600 * 

In this equation C is the cost of both loading and hauling, and 
usually includes the allowance (Item 7) for the cost, repairs 
and depreciation of the wheelbarrows, whose service is very 
short lived. Trautwine estimates this at five cents per day or 
a total of $1.05 for labor and wheelbariow. 

The number of wheelbarrow loads required for a cubic yard 
of rock, measured in place, is about twenty-four. The time 
required for loading should also be increased about one fourth; 
the time required for all purposes except hauling is therefore 
about 3.75 minutes, and the corresponding equation becomes 

Cost percubicyardof loadingand \ _CX 24(1. 25S + 3.75) 
hauling rock in wheelbarrows / 600 

(d) Scrapers. These are made in three general ways, ''buck'' 
scrapers, *'drag'' scrapers and ''wheeled " scrapers. The buck 
scraper in its original form consisted merely of a wide plank, 
shod with an iron strap on the lower edge and provided with 
a pole and a small platform on which the driver may stand to 
weight it down. The earth is not loaded on to any receptacle 
and carried, but is merely pushed over the ground. Notwith- 
standing the apparent inefficiency of the method, its extreme 
simplicity has caused its occasional adoption for the construc- 
tion of canal embankments out of material from the bed of the 
canal. The occasions are rare when their use for railroad work 
would be practicable, and even then drag scrapers would prob- 
ably be preferable. 

A drag scraper is an immense *' scoop shovel" about three feet 
long and three feet wide. There are usually two handles and a 
bail in front by which it is dragged by a team of horses. The 
nominal capacity varies from 7.5 cubic feet for the largest sizes, 
down to 3 cubic feet for the ^'one-horse" size, but these figures 
must be discounted by perhaps 40 or 50% for the actual average 
volume (as measured in the cut) loaded on during one scoop. 
The expansion of the earth during loosening is alone respons- 



§ 109. EARTHWORK. 139 

ible for a discount of 25%. These scrapers cost from $10 to 
$18. 

A wheeled' scraper is essentially an extra-large drag scraper 
which may be raised by a lever and carried on a pair of large 
wheels. Their nominal capacity ranges from 10 to 17 cubic feet, 
which should usually be liberally discounted when estimating 
output. They are loaded by dropping the scoop so that it 
scrapes up its load. The lever raises the scoop so that the load 
is carried on wheels instead of being dragged. At the dump the 
scoop is tipped so as to unload it. The movement of the 
scraper is practically continuous. They cost from $40 to 
$75. Their advantages over drag scrapers consist (1) in their 
greater capacity, (2) in the economy of transporting the load 
on wheels instead of by dragging, and (3) in the far greater 
length of haul over which the earth may be economically 
handled. 

Morris estimated the speed of drag scrapers to be 140 feet per 
minute, or 70 feet of lead per minute. The *4ead" should be 
here interpreted as the average distance from the center of the 
pit to the center of the dump. Gillette declares the speed to be 
220 feet per minute. Some of this variation may be due to dif- 
ferences in the method of measuring the distance actually trav- 
elled, especially when the lead is very short, since the scraper 
teams must always travel a considerable extra distance at each 
end in order to turn around most easily. This extra distance is 
practically constant whether the lead is long or short. Gillette 
quotes an instance w^here the length of lead was actually about 
20 feet, but the scraper teams travelled about 150 feet for each 
load carried. On this account Gillette adopts a minimum of 
75 feet of lead no matter how short the lead actually may be. 
Of course the speed depends considerably on how strictly the 
men are kept to their work and also on the care which may be 
taken to obtain a full load for each scraper. As a compromise 
between Morris's and Gillette's estimates we may adopt the con- 
venient rate of speed of 200 feet per minute, or 100 feet of lead 
per minute. There should also be allov/ed for the time lost 
in loading and unloading and for travelling the extra distance 
travelled by the teams in making the circuit. If minutes. Allow- 
ing the average value of seven loads per cubic yard and letting 
C represent the cost of scraper team and driver per day, we 
tiave for the cost as follows: 



140 RAILROAD CONSTRUCTION. § 109. 

. (73) 



Cost per cubic yard of loading and 1 _ CX7(s4-lj-) 
hauling earth in drag scrapers J 600 



In this formula C should include the cost of not only the 
driver, team, and scraper, but also the proper proportion of 
the wages of an extra man, who assists each driver in loading 
his scraper, and whose wages should be divided among the two 
(or three) scrapers to which he is assigned. Scraper work 
nearly always implies ploughing, the cost of which should be 
computed as under Item 1. 

When a low embankment is formed from borrow-pits on each 
side of the road, it may be done with scrapers, which move from 
one borrow-pit to the other, taking a load alternately from each 
side to the center and making but one half turn for each load 
carried. This reduces the time lost in turning by one third of a 
minute and reduces the constant in the numerator in Eq. (73) 
from IJ to 1. In this case the lead will usually be not greater 
than 75 feet, and therefore, if we consider this as a minimum 
value, s will ordinarily equal .75 and the quantity in the paren- 
thesis will equal 1.75. 

When using wheeled scrapers the catalogue capacity, which 
varies from 9 or 10 feet for a No. 1 scraper to 16 or 17 feet for 
a No. 3 scraper, must be reduced to 5 loads per cubic yard 
(place measurement) for a No. 1 scraper and to 2^ loads per 
cubic yard for a No. 3, not only on account of the expansion of 
the earth during loosening, but also on account of the imprac- 
ticability of loading these scrapers to their maximum nominal 
capacity. When the haul or lead for wheeled scrapers is 300 
feet or over, it will be justifiable to employ shovellers to fill up 
the bowl of the shovel, especially when the soil is tough and 
when it is impracticable to fill the shovel even approximately 
full by the ordinary method. A snatch team to assist in load- 
ing the scrapers it also economical, especially with the larger 
scrapers. The proportionate number of snatch teams to the 
total number of scrapers of course depends on the length of 
haul. The cost of these extra shovellers and extra snatch teams 
must be divided proportionally among the number of scrapers 
assisted, in determining the value C in the formula given below. 
The- extra time to be allowed on account of turning, loading, 
and dumping is about H minutes. The speed is considered 
one^station of lead per minute as before. If we call C the average 



§ 109. EARTHWORK. 141 

daily cost of one scraper and n the capacity of the scraper, or 
the number of loads per cubic yard, we may write the following 
formula: 

Cost per cubic yard of loading and 1 _ CXn(s + H) -^ ^ 

hauling earth in wheeled scrapers J 600 

(e) Cars and horses. The items of cost by this method are 
(a) charge for horses employed, (6) charge for men employed 
strictly in hauling, (c) charge for shifting rails when necessary, 
{d) repairs, depreciation, and interest on cost of cars and track. 
Part of this cost should strictly be classified under items 5 and 
7, mentioned in § 106, but it is perhaps more convenient to 
estimate them as follows: 

The traction of a car on rails is so very small that grade 
resistance constitutes a very large part of the total resistance 
if the grade is 1% or more. For all ordinary grades it is 
sufficiently accurate to say that the grade resistance is to 
the gross weight as the rise is to the distance. If the distance 
is supposed to be measured along the slope, the proportion is 
strictly true; i.e., on a 1% grade the grade resistance is 1 lb. 
per 100 of weight or 20 lbs. per ton. If the resistance on a 
level at the usual velocity is Y^^,a grade of 1:120 (0.83%) will 
exactly double it. If the material is hauled down a grade of 
1:120, the cars will run by gravity after being started. The 
work of hauling will then consist practically of hauling the 
empty cars up the grade. The grade resistance depends only 
on the rate of grade and the weight, but the tractive resistance 
will be greater per ton of weight for the unloaded than for the 
loaded cars. The tractive power of a horse is less on a grade 
than on a level, not only because the horse raises his own weight 
in addition to the load, but is anatomically less capable of 
pulling on a grade than on a level. In general it will be pos- 
sible to plan the work so that loaded cars need not be hauled up 
a grade, unless an embankment is to be formed from a low 
borrow-pit, in which case another method would probably be 
advisable. These computations are chiefly utilized in design- 
ing the method of work — the proportion of horses to cars. An 
example may be quoted from English practice (Hurst), in which 
the cars had a capacity of 3J cubic yards, weighing 30 cwt. 
empty. Two horses took five ** wagons*' J of a mile on a level 



142 RAILROAD CONSTRUCTION. § 109. 

railroad and made 15 journeys per day of 10 hours, i.e., they 
handled 250 yards per day. In addition to those on the 
"straight road,'' another horse was employed to make up 
the train of loaded wagons. With a short lead the straight- 
road horses were employed for this purpose. In the above 
example the number of men required to handle these cars, 
shift the tracks, etc., is not given, and so the exact cost of the 
above work cannot be analyzed. It may be noticed that the 
two horses travelled 22^ miles per day, drawing in one direction 
a load, including the weight of the cars, of about 57,300 lbs., 
or 28.65 net tons. Allowing -^^ as the necessary tractive 
force, it would require a pull of 477.5 lbs., or 239 lbs. for each 
horse. With a velocity of 220 feet per minute this would amount 
to H horse-power per horse, exerted for only a short time, 
however, and allowing considerable time for rest and for draw- 
ing only the empty cars. Gillette claims that the rolling re- 
sistance for such cars on a contractor's track should be con- 
sidered as 40 lbs. per ton (the equivalent of a 2% grade) and 
quotes many figures to support the assertion. Unquestionably 
the resistance on tracks with very light rails, light ties with 
wide spacing and no tamping, would be very great and might 
readily amount to 40 lbs. per ton. In the above case, the 
resistance could not have been much if any over i^-q. A re- 
sistance of 40 lbs. per ton would have required each horse to 
pull about 573 lbs. for nearly five hours per day, beside pulling 
the empty cars the rest of the time. This is far greater exertion 
than any ordinary horse can maintain. The cars generally used 
in this country have a capacity of H cubic yards and cost about 
$65 apiece. Besides the shovellers and dumping-gang, several 
men and a foreman will be required to keep the track in order 
and to make the constant shifts that are necessary. Two trains 
are generally used, one of which is loaded while the other is run 
to the dump. Some passing-place is necessary, but this is 
generally provided by having a switch at the cut and running 
the trains on each track alternately. This insures a train of 
cars always at the cut to keep the shovellers employed. The 
cost of hauling per cubic yard can only be computed when the 
number of laborers, cars, and horses employed are known, and 
these will depend on the lead, on the character of the excavation, 
on the grade, if any, etc., and must be so proportioned that the 
shovellers need not wait for cars to fill, nor the dumping-gang 



§ 109. EARTHWORK. 143 

for material to handle, nor the horses and drivers for cars to 
haul. Much skill is necessary to keep a large force in smooth 
running order. 

(f) Cars and locomotives. 30-lb. rails are the lightest that 
should be used for this work, and 35- or 40-lb. rails are better. 
One or two narrow-gauge locomotives (depending on the length 
of haul), costing about $2500 each, will be necessary to handle 
two trains of about 15 cars each, the cars having a capacity of 
about 2 cubic yards and costing about $100 each. Some cars 
can be obtained as low as $70. A force of about five men and 
a foreman will be required to shift the tracks. The track- 
shifters, except the foreman, may be common laborers. The 
dumping-gang will require about seven men. Even when the 
material is all taken doion grade the grades may be too steep for 
the safe hauling of loaded cars down the grade, or for hauling 
empty cars up the grade. Under such circumstances temporary 
trestles are necessary to reduce the grade. When these are 
used, the uprights and bracing are left in the embankment — 
only the stringers being removed. This is largely a necessity, 
but is partially compensated by the fact that the trestle forms a 
core to the embankment which prevents lateral shifting during 
settlement. The average speed of the trains may be taken as 
10 miles per hour or 5 miles of lead per hour. The time lost 
in loading and unloading is estimated (Trautwine) as 9 minutes 
or .15 of an hour. The number of trips per day of 10 hours 

^^j^ ^^j 10 ^^ 50 ^^ 

^ -i(miles of lead) + .15 (miles of lead) + .75* 

course this quotient must be a whole number. Knowing the 
number of trains and their capacity, the total number of cubic 
yards handled is known, which, divided into the total daily cost 
of the trains, will give the cost of hauling per yard. The daily 
cost of a train will include 

(a) Wages of engineer, who frequently fires his own engine; 

(6) Fuel, about J to 1 ton of bituminous coal, depending on 
work done ; 

(c) Water, a very variable item, frequently costing $3 to $5 
per day; 

{d) Repairs, variable, frequently at rate of 50 to 60% per year; 

(e) Interest on cost and depreciation, 16 to 40%. 

To these must be added, to obtain the total cost of haul, 

(/) Wages of the gang employed in shifting track. 



144 RAILROAD CONSTRUCTION. § 110. 

The above calculation for the number of train loads depends 
on the assumption that 9 minutes is total time lost by a 
locomotive for each round trip. If the haul is very short it 
may readily happen that a steam-shovel cannot fill one train 
of cars before the locomotive has returned with a load of empties 
and is ready to haul a loaded train away. The estimation of 
the number of train loads is chiefly useful in planning 
the work so as to have every tool working at its high- 
est efficiency. Usually the capacity of the steam-shovel 
or the ability to promptly "spot^' the cars under the 
shovel is the real limiting agent which determines the daily 
output. 

no. Choice of method of haul dependent on distance. In 
light side-hill work in which material need not be moved more 
than 12 or 15 feet, i.e., moved laterally across the roadbed, 
the earth may be moved most cheaply by mere shovelling. 
Beyond 12 feet scrapers are more economical. At about 100 
feet drag-scrapers and wheelbarrows are equally economical. 
Between 100 and 200 feet wheelbarrows are generally cheaper 
than either carts or drag-scrapers, but wheeled scrapers are 
always cheaper than w^heelbarrows. Beyond 500 feet two- 
wheeled carts become the most economical up to about 1700 
feet; then four-wheeled wagons become more economical up to 
3500 feet. Beyond this cars on rails, drawn by horses or by 
locomotives, become cheaper. The economy of cars on rails 
becomes evident for distances as small as 300 feet provided the 
volume of the excavation will justify the outlay. Locomotives 
• will always be cheaper than horses and mules, providing the 
work to be done is of sufficient magnitude to justify the pur- 
chase of the necessary plant and risk the loss in selling the plant 
ultimately as second-hand equipment, or keeping the plant on 
hand and idle for an indefinite period waiting for other 
work. Horses will not be economical for distances much 
over a mile. For greater distances locomotives are more 
economical, but the question of ^Mimit of profitable haul" 
(§ 116) must be closely studied, as the circumstances are 
certainly not common w^hen it is advisable to haul material 
much over a mile. 

III. Item 4. Spreading. The cost of spreading varies with 
the method employed in dumping the load. When the earth is 



§ 112. EARTHWORK. 145 

tipped over the edge of an embankment there is little if any 
necessary work. Trautwine allows about { c. per cubic yard 
for keeping the dumping-places clear and in order. This would 
represent the wages of one man at $1 per day attending to the 
unloading of 1200 two-wheeled carts each carrying J cubic yard. 
1200 carts in 10 hours would mean an average of two per minute, 
which implies more rapid and efficient work than may be de- 
pended on. The allowance is probably too small. When the 
material is dumped in layers some levelling is required, for 
which Trautwine allows 50 to 100 cubic yards as a fair day's 
work, costing from 1 to 2 cents per cubic yard. The cost of 
spreading will not ordinarily exceed this and is frequently 
nothing — all depending on the method of unloading. It should 
be noted that Mr. Morris's examples and computations (Jour. 
Franklin Inst., Sept. 1841) disregard altogether any special 
charge for this item. 

112. Item 5. Keeping Roadways in order. This feature 
is important as a measure of true economy, whatever the system 
of transportation, but it is often neglected. A petty saving in 
such matters will cost many times as much in increased labor 
in hauling and loss of time. With some methods of haul the 
cost is best combined with that of other items. 

(a) Wheelbarrows. Wheelbarrows should generally be run 
on planks laid on the ground. The adjusting and shifting of 
these planks is done by the wheelers, and the time for it is 
allowed for in the ''| minute for short rests, adjusting the 
wheeling plank, etc." The actual cost of the planks must be 
added, but it would evidently be a very small addition per cubic 
yard in a large contract. When the wheelbarrows are run on 
planks placed on ''horses" or on trestles the cost is very appre- 
ciable; but the method is frequently used with great economy. 
The variations in the requirements render any general estimate 
of such cost impracticable. 

(b) Carts and wagons. The cost of keeping roadways in 
order for carts and wagons is sometimes estimated merely as so 
much per cubic yard, but it is evidently a function of the lead. 
The work consists in draining off puddles, filling up ruts, pick- 
ing up loose stones that may have fallen off the loads, and in 
general doing everything that will reduce the traction as much 
as possible. Temporary inclines, built to avoid excessive grade 



146 RAILROAD CONSTRUCTION. § 112a. 

at some one pointy are often measures of true economy. Traut- 
wine suggests j\ c. per cubic yard per 100 feet of lead for earth- 
work and j'-^Q c. for rock work, as an estimate for this item when 
carts are used. 

(c) Cars. When cars are used a shifting-gang, consisting 
of a foreman and several men (say five), are constantly em- 
ployed in shifting the track so that the material may be loaded 
and unloaded where it is desired. The average cost of this 
item may be estimated by dividing the total daily cost of this 
gang by the number of cubic yards handled in one day. 

1 1 2a. Item 6. Trimming cuts to their proper cross- 
section. This process, often called "sand-papering," must 
be treated as an expense, since the payment received for the 
very few cubic yards of earth excavated is wholly inadequate 
to pay for the work involved. Gillette quotes bids of 2 cents 
per square yard of surface trimmed, and from this argues that, 
for average excavations, it adds to the cost four cents per cubic 
yard of the total excavation. The shallower the cut the greater 
is the proportionate cost. Of course the actual cost to the 
contractor will depend largely on the accuracy of outline de- 
manded by the engineer or inspector. 

113. Item 7. Repairs, wear, depreciation, and interest 
ON COST OF PLANT. The amount of this item evidently depends 
upon the character of the soil — the harder the soil the worse the 
wear and depreciation. The interest on cost depends on the 
current borrowing value of money. The estimate for this item 
has already been included in the allowances for horses, carts, 
ploughs, harness, wheelbarrows, steam-shovels, etc. Trautwine 
estimates | c. per cubic yard for picks and shovels. Deprecia- 
tion is generally a large percentage of the cost of earth-working 
tools, the life of all being limited to a few years, and of many 
tools to a few months. 

114. Item 8. Superintendence and incidentals. The 
incidentals include the cost of water-boys, timekeepers, watch- 
men, blacksmiths, fences, and other precautions to protect the 
public from possible injury, cost of casualty insurance for 
workmen, etc. Although the cost of some of these sub-items 
may be definitely estimated, others are so uncertain that it is 
only possible to make a lump estimate and add say 5 to 7% 
of the sum of the previous items for this item. 



§ 115. EARTHWORK. 147 

115. Contractor's profit and contingencies. The word "con- 
tingencies" here refers to the abnormal expenses caused by- 
freshets, continued wet weather, and ''hard luck/' as dis- 
tinguished from mere incidentals which are really normal 
expenses. They are the expenses which literally cannot be 
foreseen, and on which the contractor must 'Hake chances.'' 
They are therefore included with the expected profit. The 
allowance for these two elements combined is variously esti- 
mated up to 25% of the previously estimated cost of the work, 
according to the sharpness of the competition, the contractor's 
confidence in the accuracy of his estimates, and the possible un- 
certainty as to true cost owing to unfavorable circumstances. 
The contractor's real profit may vary considerably from this. 
He often pays clerks, boards and lodges the laborers in shan- 
ties built for the purpose, or keeps a supply-store, and has 
various other items both of profit and expense. His profit 
is largely dependent on skill in so handling the men that all 
can work effectively without interference or delays in wait- 
ing for others. An unusual season of bad weather will often 
affect the cost very seriously. It is a common occurrence 
to find that two contractors may be working on the same kind 
of material and under precisely similar conditions and at the 
same price, and yet one may be making money and the 
other losing it — all on account of difference of manage- 
ment. 

116. Limit of profitable haul. As intimated in §§103 and 
110, there is with every method of haul a limit of distance be- 
yond which the expense for excessive hauling will exceed the 
loss resulting from borrowing and wasting. This distance is 
somewhat dependent on local conditions, thus requiring an inde- 
pendent solution for each particular case, but the general prin- 
ciples involved will be about as follows : Assume that it has been 
determined, as in Fig. 62, that the cut and fill will exactly bal- 
ance between two points, as between e and x, assuming that, as 
indicated in § 101 (9), a trestle has been introduced between s 
and tj thus altering the mass curve to Estxn . . . Since there 
is a balance between A^ and C, the material for the fill between 
C and e' must be obtained either by "borrowing" in the im- 
mediate neighborhood or by transportation from the excavation 



148 RAILROAD CONSTRUCTION. § 116. 

between z' and n' . If cut and fill have been approximately 
balanced in the selection of grade line, as is ordinarily done, 
borrowing material for the fill C'e' implies a wastage of material 
at the cut z'n' . To compare the two methods, we may place 
against the plan of borrowing and wasting, (a) cost, if any, of 
extra right of way that may be needed from which to obtain 
earth for the fill CV; (5) cost of loosening, loading, hauling 
a distance equal to that between the centers of gravity of the 
borrow-pit and of the fill, and the other expenses incidental to 
borrowing M cubic yards for the fill CV; (c) cost of loosening, 
loading, hauling a distance equal to that between the centers 
of gravity of the cut z'n' and of the spoil-bank, and the other 
expenses incidental to wasting M cubic yards at the cut z'n^- 
(d) cost, if any, of land needed for the spoil-bank. The cost of 
the other plan will be the cost of loosening, loading, hauling (the 
hauling being represented by the trapezoidal figure Cexn), and 
the other expenses incidental to making the fill Ce' with the 
material from the cut z'n', the amount of material being M cubic 
yards, which is represented in the figure by the vertical ordi- 
nate from e to the line Cyi. The difference between these costs 
will be the cost, if any, of land for borrow-pit and spoil-bank 
plus the cost of loosening, loading, etc. (except hauling and 
roadways) of M cubic yards, minus the difference in cost of the 
excessive haul from Ce to xn and the comparatively short hauls 
from borrow-pit and to spoil-bank. 

As an illustration, taking some of the estimates previously 
given for operating with average material, the cost of all items, 
except hauling and roadways, would be about as follows: 
loosening, with plough, 1.2 c, loading 5.0 c, spreading 1.5 c, 
wear, depreciation, etc., .25 c, superintendence, etc., 1.5 c. ; 
total 8.95 c. Suppose that the haul for both borrowing and 
wasting averages 100 feet or 1 station. Then the cost of haul 
per yard, using carts, would be (§ 109, a) [125X3(1 +4)]4-600 
= 3.125 c. The cost of roadways would be about 0.1 c. per yard, 
making a total of 3.225 c. per cubic yard. Assume 3/ = 10000 
cubic yards and the area Cexn = \SOOOO yards-stations or the 
equivalent of 10000 yards hauled 1800 feet. This haul would 
cost [125X3(18 + 4)]-^600 = 13.75 c. per cubic yard. The cost 
of roadways will be 18 X .1 or 1.8 c, making a total of 15.55 c. for 
hauling and roadways. The difference of cost of hauling and 
roadways will be 15.55-(2X3.225) =9.10 c. per yard or S910 



§ 117. EARTHWORK. 149 

for the 10000 yards. Offsetting this is the cost o^ loosening, etc., 
10000 yards, at 8.95 c, costing $895. These figures may be 
better compared as follows: 



Long Haul. ■] 



fLoosening. etc., 10000 yards, (^ 8.95 c. S 895. 

! Hauling, ** 10000 " @, 15.55 c. 1555. 



$2450. 



fLoosening, etc., 10000 yards (borrowed), @ 8.95 c. S895 
I " " 10000 *' (wasted), @ 8.95 c. 895 



BoRROWixG j Hauling, etc., 10000 " (borrowed), @ 3.225 c. 322.50 

c. 322.50 

S2435.00 



Wasting '^ " " ^^^^^ " (wasted), ^ 3.225 c. 322.50 



I 



These costs are practically balanced, but no allowance has 
been made for right of way. If any considerable amount had 
to be paid for that, it would decide this particular case in favor 
of the long haul. This shows that under these conditions 1800 
feet is about the limit of profitable haul, the land costing nothing 
extra. 

BLASTING. 

117. Explosives. The effect of blasting is due to the ex- 
tremely rapid expansion of a gas which is developed by the 
decomposition of a very small amount of solid matter. Blasting 
compounds may be divided into two general classes, (a) slow- 
burning and {b) detonating. Gunpowder is a t3^pe of the slow- 
burning compounds. These are generally ignited by heat; the 
ignition proceeds from grain to grain; the heat and pressure 
produced are comparatively low. Nitro-glycerine is a type of 
the detonating compounds. They are exploded by a shock 
which instantaneously explodes the whole mass. The heat and 
pressure developed are far in excess of that produced by the 
explosion of powder. Xitro-glycerine is so easily exploded 
that it is very dangerous to handle. It was discovered that if 
the nitro-glycerine was absorbed by a spongy material like infu- 
sorial earth, it was much less liable to explode, while its power 
when actually exploded was practically equal to that of the 
amount of pure nitro-glycerine contained in the dynamite, which 
is the name given to the mixture of nitro-glycerine and infusorial 
earth. Nitro-glycerine is expensive; many other explosive 
chemical compounds which properly belong to the slow-burning 



150 RAILROAD CONSTRUCTION. § 118. 

class are comparatively cheap. It has been conclusively demon- 
strated that a mixture of nitro-glycerine and some of the cheaper 
chemicals has a greater explosive force than the sum of the 
strengths of the component parts when exploded separately. 
Whatever the reason, the fact seems established. The reason is 
possibly that the explosion of the nitro-glycerine is sufficiently 
powerful to produce a detonation of the other chemicals, which 
is impossible to produce by ordinary means, and that this explo- 
sion caused by detonation is more powerful than an ordinary 
explosion. The majority of the explosive compounds and 
"powders'' on the market are of this character — a mixture of 
20 to 60 per cent, of nitro-glycerine with variable proportions of 
one or more of a great variety of explosive chemicals. 

The choice of the explosive depends on the character of the 
rock. A hard brittle rock is most effectively blasted by a 
detonating compound. Tfie rapidity Avith which the full force 
of the explosive is developed has a shattering effect on a brittle 
substance. On the contrary, some of the softer tougher rocks 
and indurated clays are but little affected by dynamite. The 
result, is but little more than an enlargement of the blast-hole. 
Quarrying must generally be done with blasting-powder, as the 
quicker explosives are too shattering. Although the results 
obtained by various experimenters are very variable, it may be 
said that pure nitro-glycerine is eight times as powerful as black 
powder, dynamite (75% nitro-glycerine) six times, and gun- 
cotton four to six times as powerful. For open work where 
time is not particularly valuable, black powder is by far the 
cheapest, but in tunnel-headings, whose progress determines the 
progress of the whole work, dynamite is so much more effective 
and so expedites the work that its use becomes economical. 

1 1 8. Drillmg. Although many very complicated forms of 
drill-bars have been devised, the best form (with slight modifi- 
cations to suit circumstances) is as shown in Fig. 64, (a) and (b). 
The width should flare at the bottom (a) about 15 to 30%. For 
hard rock the curve of the edge should be somewhat flatter and 
for soft rock somewhat more curved than shown. Fig. 64, (a). 
Sometimes the angle of the two faces is varied from that given. 
Fig. 64, (b), and occasionally the edge is purposely blunted so 
as to give a crushing rather than a cutting effect. The drills 
will require sharpening for each 6 to 18 inches depth of hole, 
and will require a new edge to be worked every 2 to 4 days. 



§ 119. 



EARTHWORK. 



151 



For drilling vertical holes the churn-drill is the most econom- 
ical. The drill-bar is of iron, about 6 to 8 feet long, IJ" in 
diameter, weighs about 25 to 30 lbs., and is shod with a piece 
of steel welded on. The bar is lifted a few inches between each 
blow, turned partially around, and allowed to fall, the impact 
doing the work. From 5 to 15 feet of holes, depending on the 
character of the rock, is a fair day's work — 10 hours. In very 
soft rocks even more than this may be done. This method is 




Fig. 64. 



inapplicable for inclined holes or even for vertical holes in con- 
fined places, such as tunnel-headings. For such places the only 
practical hand method is to use hammers. This may be done 
by light drills and light hammers (one-man work), or by heavier 
drills held by one man and struck by one or two men with heavy 
hammers. The conclusion of an exhaustive investigation as to 
the relative economy of light or heavy hammers is that the light- 
hammer method is more economical for the softer rocks, the 
heavy-hammer method is more economical for the harder rocks, 
but that the light-hammer method is always more expeditious 
and hence to be preferred when time is important. 

The subject of machine rock-drills is too vast to be treated 
here. The method is only practicable when the amount of 
work to be done is large, and especially when time is valuable. 
The machines are generally operated by compressed air for tun- 
nel-work, thus doing the additional service of supplying fresh 
air to the tunnel-headings where it is most needed. The cost 
per foot of hole drilled is quite variable, but is usually some- 
what less than that of hand-drilling — sometimes but a small 
fraction of it. 

119. Position and direction of drill-holes. As the cost of 
drilling holes is the largest single item in the total cost of blast- 
mg, it is necessary that skill and judgment should be used in so 



152 



RAILROAD CONSTRUCTION. 



§ 120. 



locating the holes that the blasts will be most effective. The 
greatest effect of a blast will evidently be in the direction of the 
"line of least resistance." In a strictly homogeneous material 
this will be the shortest hne from the center of the explosive to 
the surface. The variations in homogeneity on account of 
laminations and seams require that each case shall be judged 
according to experience. In open-pit blasting it is generally 
easy to obtain two and sometimes three exposed faces to the 

rock, making it a simple matter 
to drill holes so that a blast will 
do effective work. When a solid 
face of rock must be broken into, 
as in a tunnel-heading, the work 
is necessarily ineffectual and ex- 
pensive. A conical or wedge- 
shaped mass will first be blown 
out by simultaneous blasts in 
the holes marked 1, Fig. 65; 
blasts in the holes marked 2 and 
3 will then complete the cross- 
section of the heading. A great saving in cost may often be 
secured by skilfully taking advantage of seams, breaks, and irreg- 
ularities. When the work is economically done there is but little 
noise or throwing of rock, a covering of old timbers and branches 
of trees generally sufficing to confine the smaller pieces which 
would otherwise fly up. 

120. Amount of explosive. The amount of explosive required 
varies as the cube of the line of least resistance. The best 
results are obtained when the line of least resistance is | of the 
depth of the hole; also when the powder fills about ^ of the hole. 
For average rock the amount of powder required is as follows : 




DRILL HOLES IN TUNNEL HEADING 

Fig. 65. 



Line of least resistance. 
Weight of powder 



2 ft. 


4 ft. 


6 ft. 


i lb. 


2 lbs. 


6^ lbs. 



8 ft. 
16 lbs. 



Strict compliance with all of the above conditions would re- 
quire that the diameter of the hole should vary for every case. 
While this is impracticable, there should evidently be some 
variation in the size of the hole, depending on the work to be 
done. For example, a 1" hole, drilled 2' 8" deep, with its 
line of least resistance 2'. and loaded ^vith ^ lb. of powder, would 



§ 121. EARTHWORK. 153 

be filled to a depth of 9^'', which is nearly J of the depth. A 
3'' hole, drilled 8' deep, with its hne of least resistanoe 6', and 
loaded with 6f lbs. of powder, would be filled to a depth of over 
28'', which is also nearly ^ of the depth. One pound of blasting- 
powder will occupy about 28 cubic inches. Quarrying necessi- 
tates the use of numerous and sometimes repeated light charges of 
powder, as a heavy blast or a powerful explosive like dynamite 
is apt to shatter the rock. This requires more powder to the 
cubic yard than blasting for mere excavation, which may usually 
be done by the use of J to |^ lb. of powder per cubic yard of easy 
open blasting. On account of the great resistance offered by 
rock when blasted in headings in tunnels, the powder used per 
cubic yard will run up to 2, 4, and even 6 lbs. per cubic yard. 
As before stated, nitro-glycerine is about eight times (and 
dynamite about six times) as powerful as the same weight of 
powder. 

121. Tamping. Blasting-powder and the slow-burning ex- 
plosives require thorough tamping. Clay is probably the best, 
but sand and fine powdered rock are also used. Wooden plugs, 
inverted expansive cones, etc., are periodically reinvented by 
enthusiastic inventors, only to be discarded for the simpler 
methods. Owing to the extreme rapidity of the development 
of the force of a nitro-glycerine or dynamite explosion, tamping 
is not so essential with these explosives, although it unquestion- 
ably adds to their effectiveness. Blasting under water has been 
effectively accomplished by merely pouring nitro-glycerine into 
the drilled holes through a tube and then exploding the charge 
without any tamping except that furnished by the superincum- 
bent water. It has been found that air-spaces about a charge 
make a material reduction in the effectiveness of the explosion. 
It is therefore necessary to carefully ram the explosive into a 
solid mass. Of course the liquid nitro-glycerine needs no ram- 
ming, but dynamite should be rammed with a wooden rammer. 
Iron should be carefully avoided in ramming gunpowder. A 
copper bar is generally used. 

122. Exploding the charge. Black powder is generally ex- 
ploded by means of a fuse which is essentially a cord in which 
there is a thin vein of gunpowder, the cord being protected by 
tar, extra linings of hemp, cotton, or even gutta-percha. The 
fuse is inserted into the middle of the charge, and the tamping 
carefully packed around it so that it will not be injured. To 



154 RAILROAD CONSTRUCTION. § 123. 

produce the detonation required to explode nitro-glycerine and 
dynamite, there must be an initial explosion of some easily 
ignited explosive. This is generally accomplished by means of 
caps containing fulminating-powder which are exploded by 
electricity. The electricity (in one class of caps) heats a very 
fine platinum wire to redness, thereby igniting the sensitive 
powder, or (in another class) a spark is caused to jump through 
the powder between the ends of two wires suitably separated. 
Dynamite can also be exploded by using a small cartridge of 
gunpowder which is itself exploded by an ordinary fuse. 

123. Cost. Trautwine estimates the cost of blasting (for 
mere excavation) as averaging 45 cents per cubic yard, falling 
as low as 30 cents for easy but brittle rock, and running up to 
60 cents and even $1 when the cutting is shallow, the rock 
especially tough, and the strata unfavorably placed. Soft tough 
rock frequently requires more powder than harder brittle rock. 

124. Classification of excavated material. The classification 
of excavated material is a fruitful source of dispute between 
contractors and railroad companies, owing mainly to the fact 
that the variation between the softest earth and the hardest rock 
is so gradual that it is very difficult to describe distinctions 
between different classifications which are unmistakable and 
indisputable. The classification frequently used is (a) earth, 
(b) loose rock, and (c) solid rock. As blasting is frequently 
used to loosen ''loose rock" and even ''earth" (if it is frozen), 
the fact that blasting is employed cannot be used as a criterion, 
especially as this would (if allowed) lead to unnecessary blasting 
for the sake of classifying material as rock. 

Earth. This includes clay, sand, gravel, loam, decomposed 
rock and slate, boulders or loose stones not greater than 1 cubic 
foot (3 cubic feet, P. R. R.), and sometimes even "hard-pan.'^ 
In general it will signify material which can be loosened by a 
plough with two horses, or with which one picker can keep one 
shoveller busy. 

Loose rock. This includes boulders and loose stones of more 
than one cubic foot and less than one cubic yard; stratified rock, 
not more than six inches thick, separated by a stratum of clay; 
also all material (not classified as earth) which may be loosened 
by pick or bar and which " can be quarried without blasting, 
although blasting may occasionally be resorted to./' 



§ 125, EARTHWORK. 155 

Solid rock includes all rock found in masses of over one cubic 
yard which cannot be removed except by blasting. 

It is generally specified that the engineer of the railroad 
company shall be the judge of the classification of the material, 
but frequently an appeal is taken from his decisions to the 
courts. 

125, Specifications for earthwork. The following specifica- 
tions, issued by the Norfolk and Western R R., represent the 
average requirements. It should be remembered that very 
strict specifications invariably increase the cost of the work, 
and frequently add to the cost more than is gained by improved 
quality of work. 

1. The grading will be estimated and paid for by the cubic 
yard, and will include clearing and grubbing, and all open ex- 
cavations, channels, and embankments required for the forma- 
tion of the roadbed, and for turnouts and sidings; cutting all 
ditches or drains about or contiguous to the road; digging the 
foundation-pits of all culverts, bridges, or walls; reconstructing 
turnpikes or common roads in cases where they are destroyed or 
interfered with; changing the course or channel of streams; and 
all other excavations or embankments connected with or incident 
to the construction of said Railroad. 

2. All grading, except where otherwise specified, whether 
for cuts or fills, will be measured in the excavations and will be 
classified under the following heads, viz.: Solid Rock, Loose 
Rock, Hard-pan, and Earth. 

Solid Rock shall include all rock occurring in masses which, 
in the judgment of the said Engineer Maintenance of Way, may 
be best removed by blasting. 

Loose Rock shall include all kinds of shale, soapstone, and 
other rock which, in the judgment of the said Engineer Main- 
tenance of Way, can be removed by pick and bar, and is soft and 
loose enough to be removed without blasting, although blasting 
may be occasionally resorted to ; also, detached stone of less than 
one (1) cubic j^ard and more than one (1) cubic foot. 

Hard-pan shall consist of tough indurated clay or cemented 
gravel, which requires blasting or other equally expensive 
means for its removal, or which cannot be ploughed with less 
than four horses and a railroad plough, or which requires two 
pickers to a shoveller, the said Engineer Maintenance of Way 
to be the judge of these conditions. 



156 RAILROAD CONSTRUCTION. § 125. 

Earth shall include all material of an earthy nature, of what- 
ever name or character, not unquestionably loose rock or hard- 
pan as above defined. 

Powder. The use of powder in cuts will not be considered 
as a reason for any other classification than earth, unless the 
material in the cut is clearly other than earth under the above 
specifications. 

3. Earth, gravel, and other materials taken from the exca- 
vations, except when otherwise directed by the said Engineer 
Maintenance of Way or his assistant, shall be deposited in the 
adjacent embankment; the cost of. removing and depositing 
which, when the distance necessary to be hauled is not more 
than sixteen hundred (1600) feet, shall be included in the price 
paid for the excavation. 

4. Extra Haul will be estimated and paid for as follows: 
whenever material from excavations is necessarily hauled a 
greater distance than sixteen hundred (1600) feet, there shall be 
paid in addition to the price of excavation the price of extra 
haul per 100 feet, or part thereof, after the first 1600 feet; the 
necessary haul to be determined in each case by the said Engi- 
neer Maintenance of Way or his assistant, from the profile and 
cross-sections, and the estimates to be in accordance therewith. 

5. All embankments shall be made in layers of such thick- 
ness and carried on in such manner as the said Engineer Mainte- 
nance of Way or his assistant may prescribe, the stone and heavy 
materials being placed in slopes and top. And in completing 
the fills to the proper grade such additional heights and fulness 
of slope shall be given them, to provide for their settlement, as 
the said Engineer Maintenance of Way, or his assistant, may 
direct. Embankments about masonry shall be built at such 
times and in such manner and of such materials as the said Engi- 
neer Maintenance of Way or his assistant may direct. 

6. In procuring materials for embankments from without 
the line of the road, and in wasting materials from cuttings, the 
place and manner of doing it shall in each case be indicated by 
the Engineer Maintenance of Way or his assistant; and care 
must be taken to injure or disfigure the land as little as possible. 
Borrow-pits and spoil-banks must be left by the Contractor in 
regular and sightly shape. 

7. The lands of the said Railroad Company shall be cleared 
to the extent required by the said Engineer Maintenance of 



§ 125. EARTHWORK. 157 

Wa}^ or his assistant, of all trees, brushes, logs, and other perish- 
able materials, which shall be destroyed by burning or deposited 
in heaps as the said Engineer Maintenance of Way, or his assist- 
ant, may direct. Large trees must be cut not more than two 
and one-half (2^) feet from the ground, and under embank- 
ments less than four (4) feet high they shall be cut close to the 
ground. All small trees and bushes shall be cut close to the 
ground. 

8. Clearing shall be estimated and paid for by the acre or 
fraction of an acre. 

9. All stumps, roots, logs, and other obstructions shall be 
grubbed out, and removed from all places where embankments 
occur less than two (2) feet in height; also, from all places where 
excavations occur and from such other places as the said Engi- 
neer Maintenance of Way or his assistant may direct. 

10. Grubbing shall be estimated and paid for by the acre or 
fraction of an acre. 

11. Contractors, when directed by the said Engineer Main- 
tenance of Way or his assistant in charge of the work, will deposit 
on the side of the road, or at such convenient points as may be 
designated, any stone, rock, or other materials that they may 
excavate; and all materials excavated and deposited as above, 
together with all timber removed from the line of the road, will 
be considered the property of the Railroad Company, and the 
Contractors upon the respective sections will be responsible for 
its safe-keeping until removed by said Railroad Company, or 
until their work is finished. 

12. Contractors will be accountable for the maintenance of 
safe and convenient places wherever public or private roads are 
in any way interfered with by them during the progress of the 
work. They will also be responsible for fences thrown down, 
and for gates and bars left open, and for all damages occasioned 
thereby. 

13. Temporary bridges and trestles, erected to facilitate the 
progress of the work, in case of delays at masonry structures 
from any cause, or for other reasons, will be at the expense of 
the Contractor. 

14. The line of road or the gradients may be changed in any 
manner, and at any time, if the said Engineer Maintenance of 
Way or his assistant shall consider such a change necessary or 
expedient; but no claim for an increase in prices of excavation 



158 RAILROAD CONSTRUCTION. § 125. 

or embankment on the part of the Contractor will be allowed 
or considered unless made in writing before the work on that 
part of the section where the alteration has been made shall have 
been commenced. The said Engineer Maintenance of Way or 
his assistant may also, on the conditions last recited, increase or 
diminish the length of any section for the purpose of more nearly 
equalizing or balancing the excavations and embankments, or 
for any other reason. 

15. The roadbed will be graded as directed by the said En- 
gineer Maintenance of Way or his assistant, and in conformity 
with such breadths, depths, and slopes of cutting and filling as 
he may prescribe from time to time, and no part of the work 
will be finally accepted until it is properly completed and dressed 
off at the required grade. 



CHAPTER IV. 

TRESTLES. 

126. Extent of use. Trestles constitute from 1 to 3% of the 
length of the average railroad. It was estimated in 1889 that 
there was then about 2400 miles of single-track railway trestle 
in the United States, divided among 150,000 structures and esti- 
mated to cost about $75,000,000. The annual charge for main- 
tenance, estimated at J of the cost, therefore amounted to about 
$9,500,000 and necessitated the annual use of perhaps 300,000,000 
ft. B. M. of timber. The corresponding figures at the present 
time must be somewhat in excess of this. The magnitude of 
this use, w^hich is causing the rapid disappearance of forests, has 
resulted in endeavors to limit the use of timber for this purpose. 
Trestles may be considered as justifiable under the following 
conditions: 

a. Permanent trestles. 

1. Those of extreme height — then called viaducts and fre- 
quently constructed of iron or steel, as the Kinzua viaduct, 302 
ft. high. 

2. Those across waterways — e.g., that across Lake Pont char- 
train, near New Orleans, 22 miles long. 

3. Those across swamps of soft deep mud, or across a river- 
bottom, liable to occasional overflow. 

h. Temporary trestles. 

1. To open the road for traffic as quickly as possible — often 
a reason of great financial importance. 

2. To quickly replace a more elaborate structure, destroyed 
by accident, on a road already in operation, so that the inter- 
ruption to traffic shall be a minimum. 

3. To form an earth embankment with earth brought from 
a distant point by the train-load, when such a measure would 
cost less than to borrow earth in the immediate neighborhood. 

4. To bridge an opening temporarily and thus allow time to 
learn the regimen of a stream in order to better proportion the 

159 



160 RAILROAD CONSTRUCTION. § 127. 

size of the waterway and also to facilitate bringing suitable stone 
for masonry from a distance. In a new country there is always 
the double danger of either building a culvert too small, requir- 
ing expensive reconstruction, perhaps after a disastrous washout, 
or else wasting money by constructing the culvert unnecessarily 
large. Much masonry has been built of a very poor quality of 
stone because it could be conveniently obtained and because 
good stone was unobtainable except at a prohibitive cost for 
transportation. Opening the road for traffic by the use of 
temporary trestles obviates both of these difficulties. 

127. Trestles vs. embankments. Low embankments are very 
much cheaper than low trestles both in first cost and mainte- 
nance. Very high embankments are very expensive to con- 
struct, but cost comparatively little to maintain. A trestle of 
equal height may cost much less to construct, but will be expen- 
sive to maintain — perhaps J of its cost per year. To determine 
the height beyond which it will be cheaper to maintain a trestle 
rather than build an embankment, it will be necessary to allow 
for the cost of maintenance. The height will also depend on 
the relative cost of timber, labor, and earthwork. At the pres- 
ent average values, it will be found that for less heights than 
25 feet the first cost of an embankment will generally be less 
than that of a trestle; this implies that a permanent trestle 
should never be constructed mth a height less than 25 feet except 
for the reasons given in § 126. The height at which a permanent 
trestle is certainly cheaper than earth^vork is more uncertain. 
A high grade line joining two hills will invariably imply at least 
a culvert if an embankment is used. If the culvert is built of 
masonry, the cost of the embankment will be so increased that 
the height at which a trestle becomes economical will be mate- 
rially reduced. The cost of an embankment increases much 
more rapidly than the height — with very high embankments 
more nearly as the square of the height — while the cost of 
trestles does not increase as rapidly as the height. Although 
local circumstances may modify the application of any set rules, 
it is probably seldom that it will be cheaper to build an embank- 
ment 40 or 50 feet high than to permanently maintain a wooden 
trestle of that height. A steel viaduct would probably be the 
best solution of such a case. These are frequently used for 
permanent structures, especially when very high. The cost of 
maintenance is much less than that of wood, which makes the 



§ 128. TRESTLES. 161 

use of iron or steel preferable for permanent trestles unless wood 
is abnormally cheap. Neither the cost nor the construction 
of iron or steel trestles will be considered in this chapter. 

128. Two principal types. There are two principal types of 
wooden trestles — pile trestles and framed trestles. The great 
objection to pile trestles is the rapid rotting of the portion of the 
pile which is underground, and the difficulty of renewal. The 
maximum height of pile trestles is about 30 feet, and even this 
height is seldom reached. Framed trestles have been con- 
structed to a height of considerably over 100 feet They are 
frequently built in such a manner that any injured piece may be 
readily taken out and renewed without interfering with traffic. 
Trestles consist of two parts — the supports called ^^ bents,'' and 
the stringers and floor system. As the stringers and floor system 
are the same for both pile and framed trestles, the "bents'' are 
all that need be considered separately. 



PILE TRESTLES. 

129. Pile bents. A pile bent consists generally of four piles 
driven into the ground deep enough to afford not only sufficient 
vertical resistance but also lateral resistance. On top of these 
piles is placed a horizontal "cap." The caps are fastened to 
the tops of the piles by methods illustrated in Fig. 66. The 
method of fastening shoAvn in each case should not be considered 
as applicable only to the particular type of pile bent used to illus- 
trate it. Fig. 66 (a and d) illustrates a mortise-joint with a hard- 
wood pin about 1^^ in diameter. The hole for the pin should 
be bored separately through the cap and the mortise, and the 
hole through the cap should be at a slightly higher level than 
that through the mortise, so that the cap will be drawn down 
tight when the pin is driven. Occasionally an iron dowel (an 
iron pin about IJ'' in diameter and about 6" long) is inserted 
partly in the cap and partly in the pile. The use of drift-bolts, 
shown in Fig. 66 (b), is cheaper in first cost, but renders repairs 
and renewals very troublesome and expensive. "Split caps," 
shown in Fig. 66 (c), are formed by bolting two half -size strips 
on each side of a tenon on top of the pile. Repairs are very 
easily and cheaply made without interference with the traffic 
and without injuring other pieces of the bent. The smaller 
pieces are more easily obtainable in a sound condition; the 



162 



RAILROAD CONSTRUCTION. 



§129. 



decay of one does not affect the other, and the first cost is but 
little if any greater than the method of using a single piece. For 
further discussion, see § 136. 

For very light traffic and for a height of about 5 feet three 
vertical piles will suffice, as sho^\Ti in Fig. 66 (a)„ Up to a height 



<^9 12*X tz'x jy f~ 

[ijl] DRIFT BOLT Ijljlll liJf 







Fig. m. 



of 8 or 10 feet four piles may be used without sway-bracing, as 
in Fig 66 (6), if the piles have a good bearing. For heights 
greater than 10 feet sway-bracing is generally necessary. The 
outside piles are frequently driven with a batter varying from 
1 : 12 to 1 : 4. 

Piles are made, if possible, from timber obtained in the 
vicinity of the ^vork. Durability is the great requisite rather 
than strength, for almost any timber is strong enough (except 
as noted below) and will be suitable if it will resist rapid decay. 
The following list is quoted as being in the order of preference 
on account of durability • 



1. Red cedar 

2. Red cypress 

3. Pitch-pine 

4. Yellow pine 



5. White pine 

6. Redwood 

7. Elm 

8. Spruce 



9. White oak 

10. Post-oak 

11. Red oak 



12. Black oak 

13. Hemlock 

14. Tamarac 



Red-cedar piles are said to have an average life of 27 years 
with a possible maximum of 50 years, but the timber is rather 



§ 130. TKESTLKS. 163 

weak, and if exposed in a river to flowing ice or driftwood is 
apt to be injured. Under these circumstances oak is prefer- 
able, although its life may be only 13 to 18 years. 

130. Methods of driving piles. The following are the prin- 
cipal methods of driving piles : 

a. A hammer weighing 2000 to 3000 lbs. or more, sliding 
in guides, is drawn up by horse-power or a portable engine, and 
allowed to fall freely. 

b. The same as above except that the hammer does not fall 
freely, but drags the rope and revolving drum as it falls and is 
thus quite materially retarded. The mechanism is a little more 
simple, but is less effective, and is sometimes made deliberately 
deceptive by a contractor by retarding the blow, in order to 
apparently indicate the requisite resistance on the part of 
the pile. 

The above methods have the advantage that the mechanism 
is cheap and can be transported into a new country with com- 
parative ease, but the w^ork done is somewhat ineffective and 
costly compared with some of the more elaborate methods 
given below. 

c. Gunpowder pile-drivers, which automatically explode a 
cartridge every time the hammer falls. The explosion not only 
forces the pile dow^n, but throws up the hammer for the next 
blow. For a given height of fall the effect is therefore doubled. 
It has been shown by experience, however, that when it is at- 
tempted to use such a pile-driver rapidly the mechanism be- 
comes so heated that the cartridges explode prematurely, and the 
method has therefore been abandoned. 

d. Steam pile-drivers, in which the hammer is operated 
directly by steam. The hammer falls freely a height of about 
40 inches and is raised again by steam. The effectiveness is 
largely due to the rapidity of the blows, which does not allow 
time between the blows for the ground to settle around the pile 
and increase the resistance, which does happen when the blows 
are infrequent. ''The hammer-cylinder weighs 5500 lbs., and 
with 60 to 75 lbs. of steam gives 75 to 80 blows per minute. 
With 41 blows a large unpointed pile was driven 35 feet into a 
hard clay bottom in half a minute. ' ' Such a driver would cost 
about $800. 

The above four methods are those usual for dry earth. In 
very soft wet or sandy soils, where an unlimited supply of water 



104 KATLROAD CONSTRUCTION. § 131. 

is available, the water-jet is sometimes employed. A pipe is 
fastened along the side of the pile and extends to the pile-point. 
If water is forced through the pipe, it loosens the sand aromid 
the point and, rising along the sides, decreases the side resist- 
ance so that the pile sinks by its own weight, aided perhaps by 
extra weights loaded on. This loading may be accomplished by 
connecting the top of the pile and the pile-driver by a block 
and tackle so that a portion of the weight of the pile-driver is 
continually thrown on the pile. 

Excessive driving frequently fractures the pile below the 
surface and thereby greatly weakens its bearing power. To 
prevent excessive '' brooming" of the top of the 
pile, owing to the action of the hammer, the top 
should be protected by an iron ring fitted to the 
top of the pile. The ^'brooming" not only ren- 
ders the driving ineffective and hence uneconomi- 
cal, but vitiates the value of any test of the bearing 
power of the pile by noting ihe sinking due to a 
given weight falling a given distance. If the pile 
is so soft that brooming is unavoidable, the top 
Fig. 67. should be adzed off frequently, and especially 
should it be done just before the final blows which are to test its 
bearing-power. 

In a new country judgment and experience will be required 
to decide intelligently whether to employ a simple drop-hammer 
machine, operated by horse-power and easily transported but 
uneconomical in operation, or a more complicated machine 
working cheaply and effectively after being transported at 
greater expense. 

131. Pile-driving formulse. If i? = the resistance of a pile, 
and s the set of the pile during the last blow, w the weight of 
the pile-hammer, and h the fall during the last blow, then we 

ivh 
may state the approximate relation that E6=wh, or R = — . 

This is the basic principle of all rational formulse, but the maxi- 
mum weight which a pile will sustain after it has been driven 
some time is by no means equal to the resistance of the pile 
during the last blow. There are also many other modifying 
elements which have been variously allowed for in the many 
proposed formulse. The formulse range from the extreme of 
empirical simplicity to very complicated attempts to allow 




§ 131. TRESTLES. 165 

properly for all modifying causes. As the simplest rule, speci- 
fications sometimes require that the piles shall be driven until 
the pile will not sink more than 5 inches under five consecutive 
blows of a 2000-lb. hammer falling 25 feet. The ^'Engineering 

News formula" "^ gives the safe load as -— j, in which w = 

weight of hammer, h=iall in feet, s = set of pile in inches under 
the last blow. This formula is derived from the above basic 
formula by calling the safe load ^ of the final resistance, and 
by adding (arbitrarily) 1 to the final set (s) as a compensation 
for the extra resistance caused by the settling of earth around 
the pile between each blow. This formula is used only for 
ordinary hammer-driving. When the piles are driven by a 

steam pile-driver the formula becomes safe load = — — . For 

^ s + 0.1 

the ^'gunpowder pile-driver/' since the explosion of the cartridge 

drives the pile in with the same force with which it throws the 

hammer upward, the effect is tiuice that of the fall of the hammer, 

4iWh/ 

and the formula becomes safe load = ^r-r. In these last two 

s + 0.1 

formulae the constant in the denominator is changed from s + 1 

to s + 0,1. The constant (1.0 or 0.1) is supposed to allow, as 

before stated, for the effect of the extra resistance caused by the 

earth settling around the pile between each blow. The more 

rapid the blows the less the opportunity to settle and the less 

the proper value of the constant. 

The above formulae have been given on account of their 
simplicity and their practical agreement with experience. Many 
other formulae have been proposed, the majority of which are 
more complicated and attempt to take into account the weight of 
the pile, resistance of the guides, etc. While these elements, 
as well as many others, have their influence, their effect is so 
overshadowed b}^ the indeterminable effect of other elements — 
as, for example, the effect of the settlement of earth around the 
pile between blows — that it is useless to attempt to employ any- 
thing but a purely empirical formula. 

Examples. 1. A pile was driven with an ordinar}^ hammer 
weighing 2500 pounds until the sinking under five consecutiA^e 
blows was 15i inches. The fall of the hammer during the last 



* Engineering News, Nov. 17, 1892. 



166 



RAILROAD CONSTRUCTION. 



§ 132. 



blows was 24 feet. What was the safe bearing power of the 
pile? 



2wh 2X2500X24 120000 
S + 1"(IX15.5) + 1'' 4.1 



= 29300 pounds. 



2. Piles are being driven into a firm soil with a steam pile- 
driver until they show a safe bearing power of 20 tons. The 
hammer weighs 5500 pounds and its fall is 40 inches. What 
should be the sinking under the final blow? 



40000 = 



5 = 



2wh 2X5500X3.33 




40000 



132. Pile- points and pile-shoes. Piles are generally sharpened 
to a blunt point. If the pile is liable to strike boulders, sunken 

logs, or other obstructions which are 
liable to turn the point, it i^ necessary 
to protect the point by some form of 
shoe. Several forms in cast iron have 
been used, also a wrought-iron shoe, 
having four '^ straps" radiating from 
the apex, the straps being nailed on to 
the pile, as shown in Fig. 68 (h). The 
cast-iron form shown in Fig. 68 (a) 
has a base cast around a drift-bolt. 
The recess on the top of the base re- 
ceives the bottom of the pile and pre- 
vents a tendency to split the bottom of the pile or to force the 
shoe off laterally. 

133. Details of design. No theoretical calculations of the 
strength of pile bents need be attempted on account of the ex- 
treme complication of the theoretical strains, the uncertainty as 
to the real strength of the timber used, the variability of that 
strength with time, and the insignificance of the economy that 
would be possible even if exact sizes could be computed. The 
piles are generally required to be not less than 10'' or 12" in 
diameter at the large end. The P. R. R. requ^es that they shall 




§ 134. TRESTLES 167 

be "not less than 14 and 7 inches in diameter at butt and small 
end respectively, exclusive of bark, which must be removed." 
The removal of the bark is generally required in good work. 
Soft durable woods, such as are mentioned in § 129, are best 
for the piles, but the caps are generally made of oak or yellow 
pine. The caps are generally 14 feet long (for single track) 
with a cross-section 12"Xl2" or 12"Xl4". "Split caps" 
would consist of two pieces 6" X 12". The sway-braces, never 
used for less heights than 6', are made of 3" X 12" timber, and 
are spiked on with |" spikes 8" long. The floor system will be 
the same as that described later for framed trestles. 

134. Cost of pile trestles. The cost, per linear foot, of pihng 
depends on the method of driving, the scarcity of suitable tim- 
ber, the price of labor, the length of the piles, and the amount 
of shifting of the pile-driver required. The cost of soft-wood 
piles varies from 8 to 15 c. per lineal foot, and the cost of oak 
piles varies from 10 to 30 c. per foot according to the length, 
the longer piles costing more per foot. The cost of driving will 
average about $2.50 per pile, or 7.5 to 10 c. per hneal foot. 
Since the cost of shifting the pile-driver is quite an item in the 
total cost, the cost of driving a long pile would be less per foot 
than for a short pile, but on the other hand the cost of the pile 
is greater per foot, which tends to make the total cost per foot 
constant. Specifications generally say that the piling will be 
paid for per lineal foot of piling left in the work. The wastage 
of the tops of piles sawed off is always something, and is fre- 
quently very large. Sometimes a small amount per foot of 
piling sawed off is allowed the contractor as compensation for 
his loss. This reduces the contractor's risk and possibly reduces 
his bid by an equal or greater amount than the extra amount 
actually paid him. 



FRAMED TRESTLES. 

135- Typical design. A typical design for a framed trestle 
bent is given in Fig. 69. This represents, with slight variations 
of detail, the plan according to which a large part of the framed 
trestle bents of the country have been built — i.e., of those less 
than 20 or 30 feet in height, not requiring multiple story con- 
struction. 

136. Joints, (a) The mortise-and-tenon joint is illustrated in 



168 RAILROAD COXSTRUCTTOX. § 136. 

Fig. 69 and also in Fig. 66 (a). The tenon should be about 





Fig. 70. 



Tic. 09. 

wide, and P>V' long. The mortise should be eut 
a little deeper than the tenon. '^Drip-holes" 
from the mortise to the outside will assist in 
draining off water that may accumulate in the 
joint and thus prevent the rapid decay that 
would otherwise ensue. These joints are very 
troublesome if a single post deca3^s and requires 
renewal. It is generally required that the mor- 
tise and tenon should be thoroughly daubed 
with paint before putting them together. This will tend to 
make the joint water-tight and prevent decay from the accu- 
mulation and retention of water in the joint. 

(b) The plaster joint. This joint is made bv bolting and 
spiking a 3''Xl2" plank on 
both sides of the joint. The 
cap and sill should be 
notched to receive the posts. 
Repairs are greath^ facili- 
tated by the use of these 
joints. This method has been 
used by the Delaware and 
Hudson Canal Co. [R. R.]. 

(c) Iron plates. An iron plate of the form shown in Fig. 72 




Fig. 71. 



§ 137. 



TRESTLES. 



169 



in Fig. 72 (a). Bolts passing 







L:'-i''''(a) 



'.'"'' 


o o 

I 


b 

c 


o 
o 




o 
o 


a 


o o 


(5) " 
a 



Fig. 72. 



apply w'ith even greater force to 



(b) is bent and used as shown 
through the bolt - holes 
shown secure the plates 
to the timbers and make 
a strong joint which may 
be readily loosened for re- 
pairs. By slight modifi- 
cations in the design the 
method may be used for 
inclined posts and compli- 
cated joints. 

(d) Split caps and sills. 
These are described in 
§ 129. Their advantages 
framed trestles. 

(e) Dowels and drift-bolts. These joints facilitate cheap and 
rapid construction, but renewals and repairs are very difficult, it 
being almost impossible to extract a drift-bolt, which has been 
driven its full length, without splitting open the pieces contain- 
ing it. Notwithstanding this objection they are extensively 
used, especially for temporary work which is not expected to 
be used long enough to need repairs. 

137. Multiple-story constnic- 
tion. Single-story framed trestle 
bents are used for heights up 
to 18 or 20 feet and exception- 
ally up to 30 feet. For greater 
heights some such construction as 
is illustrated in a skeleton design 
in Fig. 73 is used. By using split 
sills between each story and sepa- 
rate vertical and batter posts in 
each story, any piece may readily 
be removed and renewed if neces- 
sary. The height of these stories 
varies, in different designs, from 
15 to 25 and even 30 feet. In 
some designs the structure of each 
story is independent of the stories 
above and below. This greatly 
facilitates both the original construction and subsequent repairs^ 




Fig. 73. 



170 



RAILROAD CONSTRtrCTlON. 



138. 



In other designs the verticals and batter-posts are made con- 
tinuous through two consecutive stories. The structure is 
somewhat stiffer, but is much more difficult to repair. 

Since the bents of any trestle are usually of variable height 
and those heights are not always an even multiple of the uniform 
height desired for the stories, it becomes necessary to make the 




Fig. 74. 

upper stories of uniform height and let the odd amount go to the 
lowest story, as shown in Figs. 73 and 74. 

138. Span. The shorter the span the greater the number of 
trestle bents; the longer the span the greater the required strength 
of the stringers supporting the floor. Economy demands the 
adoption of a span that shall make the sum of these require- 




FiQ. 75, 



ments a minimum. The higher the trestle the greater the cost 
of each bent, and the greater the span that would be justifiable. 
Nearly all trestles have bents of variable height, but the advan- 
tage of employing uniform standard sizes is so grea^ that many 



§139. 



TRESTLES. 



171 



roads use the same span and sizes of timber not only for the 
panels of any given trestle, but also for all trestles regardless of 
height. The spans generally used vary from 10 to 16 feet. The 
Norfolk and Western R. R. uses a span of 12' 6" for all single- 
story trestles, and a span of 25' for all multiple-story trestles. 
The stringers are the same in both cases, but when the span is 
25 feet, knee-braces are run from the sill of the first story below 
to near the middle of each set of stringers. These knee-braces 
are connected at the top by a ''straining-beam" on which the 
stringers rest, thus supporting the stringer in the center and vir- 
tually reducing the span about one-half. 

139. Foundations, (a) Piles. Piles are frequently used as a 
foundation, as in Fig. 76, particularly in soft ground, and also 
for temporary structures. These 
foundations are cheap, quickly 
constructed, and are particularly 
valuable when it is financially 
necessary to open the road for 
traffic as soon as possible and 
with the least expenditure of 
money; but there is the disad- 
vantage of inevitable decay 
within a few years unless the piles are chemically treated, as will 
be discussed later. Chemical treatment, however, increases the 
cost so that such a foundation w^ould often cost more than a 
foundation of stone. A pile should be driven under each post 
as shown in Fig. 76. 

(b) Mud-sills. Fig. 




Fig. 76. 



n w w 



E 



1 SILL I 



Fig. 77. 

(c) Stone foundations. 
the most expensive. 



77 illustrates the use of mud-sills as 
built by the Louisville and 
Nashville R. R. Eight blocks 
12"X12''X6' are used under 
each bent. When the ground 
is very soft, two additional 
timbers (12" X 12" X length of 
bent-sill), as shown by the 
dotted lines, are placed under- 
neath. The number required 
evidently depends on the na- 
ture of the ground. 
Stone foundations are the best and 

For very high trestles the Norfolk and 



172 



RAILROAD CONSTRUCTION. 



§ 140 



Western R. R. employs foundations as shown in Fig. 78, the 
walls being 4 feet thick. When the height of the trestle is 72 
feet or less (the plans requiring for 72' in height a foundation- 
wall 39' 6" long) the foundation is made continuous. The sill 



SILL OF TRESTLE 






* 13 > j^ -* — 8 — *- ^ :13 *• 

Fig. 78. 



of the trestle should rest on several short lengths of 3"Xl2" 
plank; laid transverse to the sill on top of the wall. 

140. Longitudinal bracing. This is required to give the 
structure longitudinal stiffness and also to reduce the columnar 
length of the posts. This bracing generally consists of hori- 
zontal ^' waling-strips'^ and diagonal braces. Sometimes the 
braces are placed wholly on the outside posts unless the trestle 
is very high. For single-story trestles the P. R. R. employs 
the ''laced" system, i.e., a line of posts joining the cap of one 
bent with the sill of the next, and the sill of that bent with the 
cap of the next. Some plans emplo}^ braces forming an X in 
alternate panels. Connecting these braces in the center more 
than doubles their columnar strength. Diagonal braces, when 
bolted to posts, should be fastened to them as near the ends of 
the posts as possible. The sizes employed vary largely, depend- 
ing on the clear length and on whether they are expected to act 
by tension or compression. 3"Xl2" planks are often used 
when the design would require tensile strength only, and 8" X 8" 
posts are often used when compression may be expected. 

141. Lateral bracing. Several of the more recent designs of 
trestles employ diagonal lateral bracing between the caps of 
adjacent bents. It adds greatly to the stiffness of the trestle 
and better maintains its alignment. 6"X6" posts, forming 
an X and connected at the center, will answer the purpose. 

142. Abutments. When suitable stone for masonry is at 
hand and a suitable subsoil for a foundation is obtainable without 
too much excavation, a masonry abutment will be the best. 
Such an abutment would probably be used when masonry foot- 
ings for trestle bents were employed (§ 139, c). 

Another method is to construct a "crib" of 10"Xl2" timber^ 



§ 143. 



TRESTLES. 



173 



laid horizontally, drift-bolted together, securely braced and 
embedded into the ground. Except for temporary construction 
such a method is generally 
objectionable on account of 
rapid decay. 

Another method, used most 
commonly for pile trestles, and 
for framed trestles having pile 
foundations (§ 139, a), is to use 
a pile bent at such a place that 
the natural surface on the up- 
hill side is not far" below the 
cap, and the thrust of the material, filled in to bring the surface 
to grade, is insignificant. 3''Xl2'' planks are placed behind 
the piles, cap, and stringers to retain the filled material. 




Fig. 79. 



FLOOR SYSTEMS. 

14.3. Stringers. The general practice is to use two, three, 
and even four stringers under each rail. Sometimes a stringer 
is placed under each guard-rail. Generally the stringers are 
made of two panel lengths and laid so that the joints alternate. 
A few roads use stringers of only one panel length, but this prac- 
tice is strongly condemned by many engineers. The stringers 
should be separated to allow a circulation of air around them 
and prevent the decay which would occur if they were placed 
close together. This is sometimes done by means of 2'' planks, 
6' to 8' long, which are placed over each trestle bent. Several 
bolts, passing through all the stringers forming a group and 
through the separators, bind them all into one solid construc- 
tion. Cast-iron '' spools" or washers, varying from 4''' to f 
in length (or thickness), are sometimes strung on each bolt so 
as to separate the stringers. Sometimes washers are used 
between the separating planks and the stringers, the object of 
the separating planks then being to bind the stringers, especially 
abutting stringers, and increase their stiffness. 

The most common size for stringers is 8''Xl6". The Penn- 
sylvania Railroad varies the width, depth, and number of 
stringers under each rail according to the clear span. It may 
be noticed that, assuming a uniform load per running foot, both 
the pressure per square inch at the ends of the stringers (the 



174 



KAILROAD CONSTRUCTION. 



§144. 



caps having a width of 12'0 and also the stress due to trans- 
verse strain are kept approximately constant for the variable 
gross load on these varying spans. 



Clear span. 


No. of pieces 
under each rail. 


Width. 


Depth. 


10 feet 
12 •♦ 
14 " 
16 " 


2 
2 
2 
3 


8 inches 
8 " 
10 " 
8 " 


15 inches 

16 '♦ 

17 •* 
17 *• 



144. Corbels. A corbel (in trestle-work) is a stick of timber 
(perhaps two placed side by side), about 3' to 6' long, placed 
underneath and along the stringers and resting on the cap. 
There are strong prejudices for and against their use, and a 
corresponding diversity in practice. They are bolted to the 
stringers and thus stiffen the joint. They certainly reduce the 
objectionable crushing of the fibers at each end of the stringer, 
but if the corbel is no wider than the stringers, as is generally 
the case, the area of pressure between the corbels and the cap is 




FiQ. 80. 

no greater and the pressure per square inch on the cap is no less 
than the pressure on the cap if no corbels were used. If the 
corbels and cap are made of hard wood, as is recommended by 
some, the danger of crushing is lessened, but the extra cost and 
the frequent scarcity of hard wood, and also the extra cost and 
labor of using corbels, may often neutralize the advantages 
obtained by their use. 

145. Guard-rails. These are frequently made of 5''X8'' stuff, 
notched V" for each tie. The sizes vary up to 8''X8", and the 
depth of notch from f" to 1^'. They are generally bolted to 
every third or fourth tie. It is frequently specified that they 
shall be made of oak, white pine, or yellow pine. The joints 
are made over a tie, by halving each piece, as illustrated in Fig. 
81. The joints on opposite sides of the trestle should be ^'stag- 



§ 146. TRESTLES. 175 

gered." Some roads fasten every tie to the guard-rail, using a 
bolt, a spike, or a lag-screw. 

Guard-rails were originally used with the idea of preventing 
the wheels of a derailed truck from running off the ends of the 
ties. But it has been found that an outer guard-rail alone (with- 
out an inner guard-rail) becomes an actual element of danger, 
since it has frequently happened that a derailed wheel has caught 
on the outer guard-rail, thus causing the truck to slew around 




Fig. 81. 

and so produce a dangerous accident. The true function of the 
outside guard-rail is thus changed to that of a tie-spacer, which 
keeps the ties from spreading when a derailment occurs. The 
inside guard-rail generall}^ consists of an ordinary steel rail 
spiked about 10 inches inside of the running rail. These inner 
guard-rails should be bent inward to a point in the center of the 
track about 50 feet beyond the end of the bridge or trestle. If 
the inner guard-rails are placed with a clear space of 10 inches 
inside the running rail, the outer guard-rails should be at least 
6' 10" apart. They are generally much farther apart than this. 

146. Ties on trestles. If a car is derailed on a bridge or 
trestle, the heavily loaded wheels are apt to force their way be- 
tween the ties by displacing them unless the ties are closely 
spaced and fastened. The clear space between ties is generally 
equal to or less than their width. Occasionally it is a little more 
than their ^\ddth. 6''X8" ties, spaced 14" to 16" from center 
to center, are most frequently used. The length varies from 
9' to 12' for single track. They are generally notched i" deep 
on the under side where they rest on the stringers. Oak ties 
are generally required even when cheaper ties are used on the 
other sections of the road. Usually every third or fourth tie is 
bolted to the stringers. When stringers are placed underneath 
the guard-rails, bolts are run from the top of the guard-rail to 
the under side of the stringer. The guard-rails thus hold down 
the whole system of ties, and no direct fastening of the ties to 
the stringers is needed. 

147. Superelevation of the outer rail on curves. The location 
of curves on trestles should be avoided if possible, especially 
when the trestle is high. Serious additional strains are intro- 



176 



RAILROAD CONSTRUCTIOX. 



§147. 



diiced especially when the curvature is sharp or the speed high. 
Since such curves are sometimes practically unavoidable, it is 
necessary to design the trestle accordingly. If a train is stopped 
on a curved trestle, the action of the train on the trestle is 
e\'idently vertical. If the train is moving with a considerable 
velocity, the resultant of the weight and the centrifugal action 
is a force somewhat inclined from the vertical. Both of these 
conditions may be expected to exist at times. If the axis of 
the system of posts is vertical (as illustrated in methods a,h,Cjd, 
and e), any lateral force, such as would be produced by a mov- 
ing train, will tend to rack the trestle bent. If the stringers are 
set verticall}', a centrifugal force likewise tends to tip them 
sidewise. If the axis of the system of posts (or of the stringers) 
is inclined so as to coincide with the pressure of the train on the 
trestle when the train is moving at its normal velocity, there is 
no tendency to rack the trestle when the train is moving at that 
velocity, but there will be a tendency to rack the trestle or 
twist the stringers when the train is stationary. Since a moving 
train is usually the normal condition of affairs, as well as the 
condition w^hich produces the maximum stress, an inclined axis 
is evidently preferable from a theoretical standpoint; but what- 
ever design is adopted, the trestle should evidently be suffi- 
ciently cross-braced for either a moving or a stationary load, 
and any proposed design must be studied as to the effect of both 
of these conditions. Some of the various methods of securing 
the requisite superelevation may be described as follows : 

(a) Framing the outer posts longer than the inner posts, so 

that the cap is inclined at the 
proper angle; axis of posts verti- 
cal. (Fig. 82.) The method re- 
quires more work in framing the 
trestle, but simplifies subsequent 
track-laying and maintenance, im- 
less it should be found that the 
superelevation adopted is unsuit- 
able, in which case it could be cor- 
rected by one of the other methods 
given below. The stringers tend 
to twist when the train is sta- 
tionary. 

(b) Notching the cap so that the stringers are at a different 




Fig. 82. 



§ 147. 



TRESTLES. 



177 



Fig. 83. 



elevation. (Fig. 83.) This weakens the cap and requires that 
all ties shall be notched to a 
bevelled surface to fit the string- 
ers, which also weakens the ties. 
A centrifugal force will tend to 
twist the stringers and rack the 
trestle. 

(c) Placing wedges underneath 
the ties at each stringer. These 
wedges are fastened ^^-ith two 
bolts. Two or more wedges will 
be required for each tie. The ad- 
ditional number of pieces required 
for a long curve will be immense, and the work of inspection and 
keeping the nuts tight will greatly increase the cost of main- 
tenance. 

(d) Placing a wedge under the outer rail at each tie. This 
requires but one extra piece per tie. There is no need of a 
wedge under the inner tie in order to make the rail normal to 
the tread. The resulting inward inclination is substantially that 
produced by some forms of rail-chairs or tie-plates. The spikes 
(a little longer than usual) are driven through the wedge into 
the tie. Sometimes ^Mag-screws" are used instead of spikes. 
If experience proves that the superelevation is too much or too 
little, it may be changed by this method with less work than 
by any other. 

(e) Corbels of different heights. When corbels are used (see 

§ 144) the required in- 
cHnation of the floor sys- 
tem may be obtained by 
varying the depth of the 
corbels. 

(f) Tipping the whole 
trestle. This is done by 
placing the trestle on an 
inclined foundation. If 
very much inclined, the 
^. trestle bent must be se- 
cured against the possi- 
bility of slipping sidewise, 
for the slope would be considerable with a sharp curve, and the 




Fig. 84. 



178 RAILROAD COXSTRtJCTION. § 148. 

vibration of a mo\dng train would reduce the coefficient o^ 
friction to a comparatively small quantity. 

(g) Framing the outer posts longer. This case is identical 
with case (a) except that the axis of the system of posts is 
inclined, as in case (/), but the sill is horizontal. 

The above-described plans will suggest a great variety of 
methods which are possible and which differ from the above 
only in minor details. 

148. Protection from fire. Trestles are peculiarly subject to 
fire^ from passing locomotives, which may not only destro}^ the 
trestle, but perhaps cause a terrible disaster. This danger is 
sometimes reduced by placing a strip of galvanized iron along 
the top of each set of stringers and also along the tops of the 
caps. Still greater protection was given on a long trestle on the 
Louisville and Nashville R. R. by making a solid flooring of 
timber, covered with a layer of ballast on which the ties and 
rails were laid as usual. 

Barrels of water should be provided and kept near all trestles, 
and on very long trestles barrels of water should be placed every 
two or three hundred feet along its length. A place for the bar- 
rels may be provided by using a few ties which have an extra 
length of about four feet, thus forming a small platform, which 
should be surrounded by a railing. The track-walker should be 
held accountable for the maintenance of a supply of water in 
these barrels, renewals being frequently necessary on account of 
evaporation. Such platforms should also be provided as refuge- 
bays for track-walkers and trackmen working on the trestle. On 
very long trestles such a platform is sometimes pro\aded with 
sufficient capacity for a hand-car. 

149. Timber. Any strong durable timber may be used when 
the choice is limited, but oak, pine, or cypress are preferred 
when obtainable. When all of these are readily obtainable, 
the various parts of the trestle will be constructed of different 
kinds of wood — the stringers of long-leaf pine, the posts and 
braces of pine or red cypress, and the caps, sills, and corbels (if 
used) of white oak. The use of oak (or a similar hard wood) 
for caps, sills, and corbels is desirable because of its greater 
strength in resisting crushing across the grain, which is the 
critical test for these parts. There is no physiological basis to 
the objection, sometimes made, that different species of timber, 
iSk contact with each other, will rot quicker than if only one 




(7*0 face page 178.) 



§ 150. TRESTLES. 179 

kind of timber is used. When a very extensive trestle is to be 
built at a place where suitable growing timber is at hand but 
there is no convenient sawmill, it will pay to transport a port- 
able sawmill and engine and cut up the timber as desired. 

150. Cost of framed timber trestles. The cost varies widely 
on account of the great variation in the cost of timber. When 
a railroad is first penetrating a new and undeveloped region, the 
cost of timber is frequently small, and when it is obtainable from 
the company's right-of-way the only expense is felling and 
sawing. The work per M, B. M., is small, considering that a 
single stick 12" X 12" X 25' contains 300 feet, B. M., and that 
sometimes a few hours' work, worth less than $1, will finish all 
the work required on it. Smaller pieces will of course require 
more w^ork per foot, B. M. Long-leaf pine can be purchased 
from the mills at from $8 to $12 per M feet, B. M., according 
to the dimensions. To this must be added the freight and labor 
of erection. The cartage from the nearest railroad to the trestle 
may often be a considerable item. Wrought iron will cost 
about 3 c. per pound and cast iron 2 c., although the prices are 
often lower than these. The amount of iron used depends on 
the detailed design, but, as an average, will amount to $1.50 
to $2 per 1000 feet, B. M., of timber. A large part of the tres- 
tling of the country has been built at a contract price of about 
$30 per 1000 feet, B. M., erected. While the cost will frequently 
rise to $40 and even $50 when timber is scarce, it will drop to 
$13 (cost quoted) when timber is cheap. 



DESIGN OF WOODEN TRESTLES. 

151. Common practice. A great deal of trestling has been 

constructed without any rational design except that custom and 
experience have shown that certain sizes and designs are probably 
safe. This method has resulted occasionally in failures but more 
frequently in a very large waste of timber. Many railroads 
employ a uniform size for all posts, caps, and sills, and a uniform 
size for stringers, all regardless of the height or span of the 
trestle. For repair work there are practical reasons favoring 
this. ''To attempt to run a large lot of sizes would be more 
wasteful in the end than to maintain a few stock sizes only. 
Lumber can be bought more cheaply by giving a general order 
for ' the run of the mill for the season,' or ' a cargo lot,' specify 



180 RAILROAD CONSTRUCTION. § 152. 

ing approximate percentages of standard stringer size, of 
12 X 12-inch stuffy 10 X 10-inch stuff, etc., and a Hberal propor- 
tion of 3- or 4-inch plank, all lengths thro^Mi in. The 12 X 12- 
inch stuff, etc., is ordered all lengths, from a certain specified 
length up. In case of a wreck, washout, burn-out, or sudden 
call for a trestle to be completed in a stated time, it is much 
more economical and practical to order a certain number of 
carloads of Hrestle stuff' to the ground and there to select piece 
after piece as fast as needed, dependent only upon the length of 
stick required. WTien there is time to make the necessary sur- 
veys of the ground and calculations of strength, and to wait for a 
special bill of timber to be cut and delivered, the use of differ- 
ent sizes for posts in a structure would be warranted to a certain 
extent." * For new construction, when there is generally 
sufficient time to design and order the proper sizes, such waste- 
fulness is less excusable, and under an}^ conditions it is both 
safer and more economical to prepare standard designs which 
can be made applicable to varying conditions and which will at 
the same time utilize as much of the strength of the timber as 
can be depended on. In the following sections will be given 
the elements of the preparation of such standard designs, which 
will utilize uniform sizes wath as little waste of timber as possible. 
It is not to be understood that special designs should be made 
for each individual trestle. 

152. Required elements of strength. The stringers of trestles 
are subject to transverse strains, to crushing across the* grain 
at the ends, and to shearing along the neutral axis. The strength 
of the timber must therefore be computed for all these kinds 
of stress. Caps and sills will fail, if at all, by crushing across 
the grain; although subject to other forms of stress, these could 
hardly cause failure in the sizes usually employed. There is an 
apparent exception to this: if piles are improperly driven and 
an uneven settlement subsequently occurs, it may have the 
effect of transferring practically all of the weight to two or three 
piles, while the cap is subjected to a severe transverse strain 
which may cause its failure. Since such action is caused gener- 
ally by avoidable errors of construction it may be considered as 
abnormal, and since such a failure will generally occur by a 
gradual settlement, all danger may be avoided by reasonable 

* From "Economical Designing of Timber Trestle Bridges." 



§ 153. TRESTLES. 181 

care in inspection. Posts must be tested for their columnar 
strength. These parts form the bulk of the trestle and are the 
parts which can be definitely designed from known stresses. 
The stresses in the bracing are more indefinite, depending on 
indeterminate forces, since the inclined posts take up an un- 
known proportion of the lateral stresses, and the design of the 
bracing may be left to what experience has shown to be safe, 
without involving any large waste of timber. 

153. Strength of timber. Until recently tests of the strength 
of timber have generally been made by testing small, selected, 
well-seasoned sticks of ^^ clear stuff," free from knots or imper- 
fections. Such tests would give results so much higher than 
the vaguely known strength of large unseasoned ^^ commercial" 
timber that very large factors of safety were recommended — • 
factors so large as to detract from any confidence in the whole 
theoretical design. Recently the U. S. Government has been 
making a thoroughly scientific test of the strength of full-size 
timber under various conditions as to seasoning, etc. The work 
has been so extensive and thorough as to render possible the 
economical designing of timber structures. 

One important result of the investigation is the determina- 
tion of the great influence of the moisture in the timber and 
the law of its efTect on the strength. It has been also shown 
that timber soaked T\dth water has substantially the same 
strength as green timber, even though the timber had once been 
thoroughly seasoned. Since trestles are exposed to the weather 
they should be designed on the basis of using green timber. 
It has been shown that the strength of green timber is very 
regularly about 55 to 60% of the strength of timber in which 
the moisture is 12% of the dry weight, 12% being the proportion 
of moisture usually found in timber that is protected from the 
weather but not heated, as, e.gr., the timber in a barn. Since 
the moduli of rupture have all been reduced to this standard of 
moisture (12%), if we take one-eighth of the rupture values, it 
still allows a factor of safety of about five, even on green timber. 
On page 172 there are quoted the values taken from the U. S. 
Government reports on the strength of timber, the tests probably 
being the most thorough and reliable that were ever made. 

On page 173 are given the ^^ average safe allowable working 
unit stresses in pounds per square inch," as recommended by 
the committee on '" Strength of Bridge and Trestle Timbers," 



182 



RAILROAD CONSTRUCTION. 



§154 



the work being done under the auspices of the Association of 
Railway Superintendents of Bridges and Buildings. The report 
was presented at their fifth annual convention, held in New 
Orleans, in October, 1895. 

Moduli of rupture for various timbers. [12% moisture.] 
(Condensed from U. S. Forestry Circular, No. 15.) 









U o 


Cross-bending. 


Crush- 
ing 
end- 
wise. 


o 


bO 

o . 
an 


No. 


Species. 


It 

eg 


Modulus 

of 
Elasticity. 


1 
2 
3 
4 
5 
6 
7 


Long-leaf nine. . . . 
Cuban " .... 

Short-leaf " 

Loblolly " .... 
White " .... 
Red " .... 
Spruce " .... 


38 
39 
32 
33 
24 
31 
39 


12 600 

13 600 

10 100 

11 300 
7 900 
9 100 

10 000 


2 070 000 
2 370 000 

1 680 000 

2 050 000 
1 390 000 
1 620 000 
1 640 000 


8000 
8700 
6500 
7400 
5400 
6700 
7300 


1180 
1220 

960 
1150 

700 
1000 
1200 


700 
700 
700 
700 
400 
500 
800 


8 


Bald cypress 


29 
23 
32 


7 900 

6 300 

7 900 


1 290 000 

910 000 

1 680 000 


6000 
5200 
5700 


800 
700 
800 


500 


9 

10 


White cedar 

Douglas soruce.. . . 


400 
500 


1 1 


White oak 


50 
46 
50 
46 
45 
46 
45 
46 


13 100 

11 300 

12 300 
11 500 

11 400 

13 100 
10 400 

12 000 


2 090 000 

1 620 000 

2 030 000 
1 610 000 
1 970 000 
1 860 000 
1 750 000 
1 930 000 


8500 i 2200 
7300 1900 
7100 ' 3000 


1000 


\o 


Overcup " . . 




1000 


13 


Post " . . 




1100 


14 


Cow " . 




7400 
7200 
8100 
7200 
7700 


1900 
2300 
2000 
1600 
1800 


900 


1 5 


Red '* . . 




1100 


16 


Texan " . . 




900 


19 


Willow " . . 




900 


*>0 


Spanish " 




900 








21 
27 


Shagbark hickory. . 
Pignut " 
White elm 


51 
56 
34 
46 
39 


16 000 
18 700 
10 300 
13 500 
10 800 


2 390 000 
2 730 000 
1 540 000 
1 700 000 
1 640 000 


9500 
10900 
6500 
8000 
7200 


2700 
3200 
1200 
2100 
1900 


1100 

1200 

800 


9q 


Cedar " 


1300 


30 


White ash 


1100 











154. Loading. As shown in § 138, the span of trestles is always 
small, is generally 14 feet, and is never greater than 18 feet 
except w^hen supported by knee-braces. The greatest load that 
wdll ever come on any one span will be the concentrated loading 
of the drivers of a consolidation locomotive. With spans of 14 
feet or less it is impossible for even the four pairs of drivers to 
be on the same span at once. The weight of the rails, ties, and 
guard-rails should be added to obtain the total load on the string- 
ers, and the w^eight of these, plus the weight of the stringers, 
should be added to obtain the pressure on the caps or corbels. 



§154. 



TRESTLES. 



183 







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184 



RAILROAD CONSTRUCTION. 



§155. 



This dead load is almost insignificant compared with the live 
load and may be included with it. The weight of rails, ties, 
etc., may be estimated at 240 pounds per foot. To obtain the 
weight on the caps the weight of the stringers must be added, 
which depends on the design and on the weight per cubic foot 
of the wood employed. But as the weight of the stringers is 
comparatively small, a considerable percentage of variation in 
weight will have but an insignificant effect on the result. Dis- 
regarding all refinements as to actual dimensions, the ordinary 
maximum loading for standard-gauge railroads may be taken 
as that due to four driving-axles, spaced 5' 0'' apart and giving 
a pressure of 40000 pounds per axle. This should be increased 
to 54000 pounds per axle (same spacing) for the heaviest traffic. 
On the basis of 40000 pounds per axle or 20000 pounds per wheel 
the following results have been computed: This loading is 
assumed to allow for impact. 

STRESSES ON VARIOUS SPANS DUE TO MOVING LOADS OF 20000 
POUNDS, SPACED 5' 0" APART, WITH 120 POUNDS PER FOOT 
OF LIVE LOAD. 



Span in feet. 


Max. moment, 
ft. lbs. 


Max. shear. 


Max. load on 

one cap under 

one rail. 


10 
12 
14 
16 
18 


51 500 

82 160 

112 940 

123 840 

164 860 


30 600 
35 720 
39 410 
43 460 
47 747 


41 200 
49 440 
57 680 
65 920 
75 160 



Although the dead load does not vary in proportion to the 
live load, yet, considering the very small influence of the dead 
load, there will be no appreciable error in assuming the corre- 
sponding values, for a load of 54000 lbs. per axle, to be ^^ of 
those given in the above tabulation. 

155. Factors of safety. The most valuable result of the gov- 
ernment tests is the knowledge that under given moisture condi- 
tions the strength of various species of sound timber is not the 
variable uncertain quantity it was once supposed to be, but that 
its strength can be relied on to a comparatively close percentage. 
This confidence in values permits the employment of lower fac- 
tors of safety than have heretofore been permissible. Stresses, 
which when excessive would result in immediate destruction, 
such as cross-breaking and columnar stresses, should be allowed 
a higher factor of safety — say 6 or 8 for green timber. Other 
stresses, such as crushing across the grain and shearing along the 



§ 156. TRESTLES. 1S5 

neutral axis, which will be apparent to inspection before it is 
dangerous, may be allowed lower factors — sa}^ 3 to 5. 

156. Design of stringers. The strength of rectangular beams 
of equal width varies as the square of the depth; therefore deep 
beams are the strongest. On the other hand, when any cross- 
sectional dimension of timber much exceeds 12" the cost is 
much higher per M, B. M., and it is correspondingly difficult to 
obtain thoroughly sound sticks, free from wind-shakes, etc. 
Wind-shakes especially affect the shearing strength. Also, if 
the required transverse strength is obtained by using high nar- 
row stringers, the area of pressure between the stringers and the 
cap may become so small as to induce crushing across the grain. 
This is a very common defect in trestle design. As already in- 
dicated in § 138, the span should vary roughly with the average 
height of the trestle, the longer spans being employed when the 
trestle bents are very high, although it is usual to employ the 
same span throughout any one trestle. 

To illustrate, if we select a span of 14 feet, the load on one 
cap will be 57680 lbs. If the stringers and cap are made of 
long-leaf yellow pine, the allowable value, according to the table 
on page 183 for "compression across the grain '' is 350 pounds 
per square inch; this will require 165 square inches of surface. 
If the cap is 12'' wide, this will require a width of 14 inches, or 
say 2 stringers under each rail, each 7 inches wide. For rectan- 
gular beams 

Moment = ^R'hh^. 

Using for R^ the safe value 1200 lbs. per square inch, we have 

112940Xl2 = iXl200Xl4X/l^ 
from which h = 22'\0. If desired, the width may be increased 
to 10" and the depth correspondingly reduced, which will give 
similarly h= 18".4 or say 18^''. This shows that two beams, 
10"X 18^'', under each rail will stand the transverse bending and 
have more than enough area for crushing. 
The shear per square inch will equal 

3 total shear 3 39410 

2 cross-section = 2" 2X10X18^ = ^^^ ^^^' ^^' "^- ^^^^^' 

which is allowable, although it should preferably be less. Hence 
the above combination of dimensions will answer. This is a 
deeper beam than is called for by the tabular form on page 174, 
but is evidently based on heavier loading than was commonly 
used when those dimensions were adopted. 



186 RAILROAD CONSTRUCTION. § 157. 

The deflection should be computed to see if it exceeds the 
somewhat arbitrary standard of -j^^jj of the span. The deflec- 
tion for uniform loading is 

, 5WP 



'32bh'E ' 



in which Z= length in inches; 

"pr= total load, assumed as uniform; 

E= modulus of elasticity, given as 850,000 lbs. 

per sq. in. for long-leaf pine, according to the table on p. 183. 
Then 

5X57680X1683 ^ 
32X20X18.53X850000 ^ ^ 



2JqX168"=0".84, 



so that the calculated deflection is well within the limit. Of 
course the holding is not strictly uniform, but even with a lib- 
eral allowance the deflection is still safe. 

For the heaviest practice (54000 lbs. per axle) these stringer 
dimensions must be correspondingly increased. 

157. Design of posts. Four posts are generally used for 
single-track work. The inner posts are usually braced by the 
cross-braces, so that their columnar strength is largely increased; 
but as they are apt to get more than their share of work, the ad- 
vantage is compensated and they should be treated as unsup- 
ported columns for the total distance between cap and sill in 
simple bents, or for the height of stories in multiple-story con- 
struction. The caps and sills are assumed to have a width of 12''. 
It facilitates the application of bracing to have the columns of 
the same width and vary the other dimension as required. 

Unfortunately the experimental work of the U. S. Govern- 
ment on timber testing has not yet progressed far enough to 
establish unquestionably a general relation between the strength 
of long columns and the crushing strength of short blocks. The 



§ 157. TRESTLES. 187 

following formula has been suggested, but it cannot be consid- 
ered as established: 

, z.., 700 + 15c . , . , 

/=^X700T15^T7- '""^^'^ 

/= allowable working stress per sq. in for long columns; 
F= '' '' '' '' '' '' '' short blocks; 

I 

Z = length of column in inches; 

d= least cross-sectional dimensions in inches. 

Enough work has been done to give great reliability to the two 
following formulae for white pine and yellow pine, quoted from 
Johnson's ^'Materials of Construction," p. 684: 

1 /Z\ 2 
Working load per sq. in. = ;? = 1000 "" 7" ( 7~ ) ' long-leaf pine ; 

1 / ^ \ ^ 
=p = 600 — ^ I T- 1 ; white pine; 



ci CI ( ( a II 



in which Z= length of column in inches, ai 

/i = least cross-sectional dimension in inches. 

The frequent practice is to use 12"Xl2" posts for all trestles. 
If we substitute in the above formula Z = 20' =240" and h = l2" 
we have p = 1000 -KV/)^ =900 lbs. 

900X144 = 129600 lbs., the working load for each post. This 
is more than the total load on one trestle bent and illustrates 
the usual great waste of timber. Making the post 8" X 12" and 
calculating similarly, we have p = 775, and the working load per 
column is 775X96 = 74400 lbs. As considerable must be 
allowed for 'Sveathering," which destroys the strength of the 
outer layers of the wood, and also for the dynamic effect of 
the live load, 8"Xl2" may not be too great, but it is certainly 
a safe dimension. 12" X 6" would possibly prove amply safe 
in practice. One method of allowing for weathering is to dis- 
regard the outer half-inch on all sides of the post, i.e., to cal- 
culate the strength of a post one inch smaller in each dimension 
than the post actually employed. On this basis an 8" X 12" X20' 



188 RAILROAD COXSTRUCTIOX. § 158. 

post, computed as a 7"Xll' post, would have a safe columnar 
strength of 706 lbs. per square inch. With an area of 77 square 
inches, this gives a working load of 54362 lbs. for each post, or 
217448 lbs. for the four posts. Considering that 74200 lbs. 
is the maximum load on one cap (14 feet span), the great excess 
of strength is apparent. 

158. Design of caps and sills. The stresses in caps and sills 
are very indefinite, except as to crushing across the grain. As 
the stringers are placed almost directly over the inner posts, and 
as the sills are supported just under the posts, the transverse 
stresses are almost insignificant. In the abov^e case four posts 
have an area of 4 X 12'' X 8'' =384 sq. in. The total load, 
74200 lbs.; will then give a pressure of 193 pounds per square 
inch, which is within the allowable limit. This one feature 
might require the use of 8"Xl2'' posts rather than 6''Xl2" 
posts, for the smaller posts, although probably strong enough as 
posts, would produce an objectionably high pressure. 

159. Bracing. Although some idea of the stresses in the 
bracing could be found from certain assumptions as to wind- 
pressure, etc., yet it would probably not be found wise to de- 
crease, for the sake of economy, the dimensions which practice 
has shown to be sufiiicient for the work. The economy that 
would be possible would be too insignificant to justify any risk. 
Therefore the usual dimensions, given in §§ 139 and 140, should 
be employed. 



CHAPTER V. 

TUNNELS. 
SURVEYING. 

1 60. Surface surveys. As tunnels are always dug from each 
end and frequently from one or more intermediate shafts, it is 
necessary that an accurate surface survey should be made 
between the two ends. As the natural surface in a locality 
where a tunnel is necessary is almost invariably yerj steep and 
rough, it requires the employment of unusually refined methods 
of work to avoid inaccuracies. It is usual to run a line on the 
surface that will be at every point vertically over the center line 
of the tunnel. Tunnels are generally made straight unless 
curves are absolutely necessary, as curves add greatly to the 
cost. Fig. 85 represents roughly a longitudinal section of the 




Fig. 



— -H*----7O0O- *|*--6000- — *j--— 7000--— *j 6000 n*—- 5000- 

85. — Sketch of Section of the Hoosac Tunnel. 



Hoosac Tunnel. Permanent stations were located at A, B, C, 
Dj Ey and F, and stone houses were built at A, Bj C, and D. 
These were located with ordinary field transits at first, and then 
all the points were placed as nearly as possible in one vertical 
plane by repeated trials and minute corrections, using a very 
large specially constructed transit. The stations D and F were 
necessary because E and A were invisible from C and B, The 
alignment at A and E having been determined with great accu- 
racy, the true alignment was easily carried into the tunnel. 

189 



190 BAILROAD CONSTRUCTION. § 161. 

The relative elevations of A and E were determined with 
great accuracy. Steep slopes render necessary many settings 
of the level per unit of horizontal distance and require that the 
work be unusually accurate to obtain even fair accuracy per 
unit of distance. The levels are usually re -run many times 
until the probable error is a very small quantity 

The exact horizontal distance between the two ends of the 
tunnel must also be known, especially if the tunnel is on a 
grade. The usual steep slopes and rough topography likewise 
lender accurate horizontal measurements very difficult. Fre- 
quently when the slope is steep the measurement is best ob- 
tained by measuring along the slope and allowing for grade. 
This may be very accurately done by employing two tripods 
(level or transit tripods serve the purpose very well), setting 
them up slightly less than one tape-length apart and measuring 
between horizontal needles set in wooden blocks inserted in the 
top of each tripod. The elevation of each needle is also observed. 
The true horizontal distance between two successive positions 
of the needles then equals the square root of the difference of 
the squares of the inclined distance and the difference of eleva- 
tion. Such measurements will probably be more accurate than 
those made by attempting to hold the tape horizontal and 
plumbing down with plumb-bobs, because (1) it is practically 
difficult to hold both ends of the tape truly horizontal; (2) on 
steep slopes it is impossible to hold the down-hill end of a 100- 
foot tape (or even a 25-foot length) on a level with the other 
end, and the great increase in the number of applications of the 
unit of measurement very greatly increases the probable error 
of the whole measurement; (3) the vibrations of a plumb-bob 
introduce a large probability of error in transferring the meas- 
urement from the elevated end of the tape to the ground, and 
the increased number of such applications of the unit of meas- 
urement still further increases the probable error. 

i6i. Surveying down a shaft. If a shaft is sunk, as at S^ 
Fig 85, and it is desired to dig out the tunnel in both directions 
from the foot of the shaft so as to meet the headings from the 
outside, it is necessary to know, when at the bottom of the 
shaft, the elevation, alignment, and horizontal distance from 
each end of the tunnel. 

The elevation is generally carried down a shaft by means of 
a steel tape. This method involves the least number of appli- 



§ 161. TUNNELS. 191 

cations of the unit of measurement and greatly increases the 
accuracy of the final result. 

The horizontal distance from each end may be easily trans- 
ferred do-^vn the shaft by means of a plumb-bob, using some of 
the precautions described in the next paragraph. 

To transfer the alignment from the surface to the bottom of 
a shaft requires the highest skill because the shaft is always 
small, and to produce a line perhaps several thousand feet long 
in a direction given by two points 6 or 8 feet apart requires 
that the two points must be determined with extreme accuracy. 
The eminently successful method adopted in the Hoosac Tunnel 
will be briefly described: Two beams were securely fastened 
across the top of the shaft (1030 feet deep), the beams being 
placed transversely to the direction of the tunnel and as far 
apart as possible and yet allow plumb-lines, hung from the 
intersection of each beam with the tunnel center line, to swing 
freely at the bottom of the shaft. These intersections of the 
beams with the center line were determined by averaging the 
results of a large number of careful observations for alignment. 
Two fine parallel wires, spaced about ^^'^ apart, were then 
stretched between the beams so that the center line of the 
tunnel bisected at all points the space between the wires. 
Plumb-bobs, weighing 15 pounds, were suspended by fine wires 
beside each cross-beam, the wires passing between the two 
parallel alignment wires and bisecting the space. The plumb- 
bobs were allowed to swing in pails of w^ater at the bottom. 
Drafts of air up the shaft required the construction of boxes 
surrounding the wires. Even these precautions did not suffice 
to absolutely prevent vibration of the wire at the bottom 
through a very small arc. The mean point of these vibrations 
in each case was then located on a rigid cross-beam suitably 
placed at the bottom of the shaft and at about the level of the 
roof of the tunnel. Short plumb-lines were then suspended 
from these points whenever desired; a transit was set (by trial) 
so that its line of collimation passed through both plumb-lines 
and the line at the bottom could thus be prolonged. 

Some recent experience in the ^'Tamarack" shaft, 4250 feet 
deep, shows that the accuracy of the results may be affected by 
air-currents to an unsuspected extent. Two 50-lb. cast-iron 
plumb-bobs were suspended with No. 24 piano-wire in this 
shaft. The carefully measured distances between the wires 



192 



RAILROAD CONSTRUCTION. 



§162. 



at top and bottom were 16.32 and 16.43 feet respectively. 
After considerable experimenting to determine the cause of 
the variation, it was finally concluded that air-currents were 
alone responsible. The variation of the bobs from a true ver- 
tical plane passing through the wires at the top was of course 
an unknown quantity, but since the variation in one direction 
amounted to 0.11 foot, the accuracy in other directions was 
very questionable. This shows that a careful comparative 
measurement between the wires at top and bottom should 
always be made as a test of their parallelism. 

162. Underground surveys. Survey marks are frequently 
placed on the timbering, but they are apt to prove unreliable 
on account of the shifting of the timbering due to settlement 
of the surrounding material. They should never be placed at 
the bottom of the tunnel on account of the danger of being 
disturbed or covered up. Frequently holes are drilled in the 
roof and filled with wooden plugs in which a hook is screwed 
exactly on line Although this is probably the safest method, 
even these plugs are not always undisturbed, as the material, 
unless very hard, will often settle slightly as the excavation 
proceeds. When a tunnel is perfectly straight and not too long, 
alignment-points may be given as frequently as desired from 

permanent stations located outside 
the tunnel where they are not liable 
to disturbance. This has been ac- 
complished by running the align- 
ment through the upper part of the 
cross-section, at one side of the cen- 
ter, where it is out of the way of 
the piles of masonry material, 
debris, etc., w^hich are so apt to 
choke up the lower part of the 
cross-section. The position of this 
line relative to the cross-section 
being fixed, the alignment of any 
required point of the cross-section 
is readily found by means of a light 
frame or template with a fixed tar- 
get located where this line would intersect the frame when 
properly placed. A level-bubble on the frame will assist in 
setting the frr.me in its proper position. 




§ 163. TUNNELS. 193 

In all tunnel surveying the cross-wires- must be illuminated 
by a lantern, and the object sighted at must also be illuminated. 
A powerful dark-lantern with the opening covered with ground 
glass has been found useful. This may be used to illuminate a 
plumb-bob string or a very fine rod, or to place behind a brass 
plate having a narrow slit in it, the axis of the slit and plate 
being coincident with the plumb-bob string by which it is 
hung. 

On account of the interference to the surveying caused by 
the work of construction and also by the smoke and dust in the 
air resulting from the blasting, it is generally necessary to make 
the surveys at times when construction is temporarily sus- 
pended. 

163. Accuracy of tunnel surveying. Apart from the very 
natural desire to do surveying which shall check well, there is 
an important financial side to accurate tunnel surve3dng. If 
the survey lines do not meet as desired when the headings come 
together, it may be found necessary, if the error is of appreciable 
size, to introduce a slight curve, perhaps even a reversed curve, 
into the alignment, and it is even conceivable that the tunnel 
section would need to be enlarged somewhat to allow for these 
curves. The cost of these changes and the perpetual annoyance 
due to an enforced and undesirable alteration of the original 
design will justify a considerable increase in the expenses of the 
survey. Considering that the cost of surveys is usually but a 
small fraction of the total cost of the work, an increase of 10 or 
even 20% in the cost of the surveys will mean an insignificant 
addition to the total cost and frequent /, if not generally, it will 
result in a saving of many times the increased cost. The 
accuracy actually attained in two noted American tunnels is 
given as follows: The Musconetcong tunnel is about 5000 feet 
long, bored through a mountain 400 feet high. The error of 
alignment at the meeting of the headings was 0'.04, error of 
levels O'.Olo, error of distance 0'.52. The Hoosac tunnel is 
over 25000 feet long. The heading from the east end met the 
heading from the central shaft at a point 11274 feet from the 
east end and 1563 feet from the shaft. The error in alignment 
was ^^6 of an inch, that of levels "a few hundredths," error of 
distance ^'trifling." The alignment, corrected at the shaft, 
was carried on through and met the heading from the west end 
at a point 10138 feet from the west end and 2056 feet from 



194 RAILROAD CONSTRUCTION. § 164. 

the shaft. Here the error of alignment was y^g'^ and that of 
levels 0.134 ft. 

DESIGN. 

164. Cross-sections, Nearly all tunnels have cross-sections 
peculiar to themselves — all varying at least in the details. The 
general form of a great many tunnels is that of a rectangle sur- 
mounted by a semi-circle or semi-ellipse. In very soft material 
an inverted arch is necessary along the bottom. In such cases 
the sides will generally be arched instead of vertical. The sides 
are frequently battered. With very long tunnels, several forms 
of cross-section will often be used in the same tunnel, owing to 
differences in the material encountered. In solid rock, which 
will not disintegrate upon exposure, no lining is required, and 
the cross-section will be the irregular section left by the blasting, 
the only requirement being that no rock shall be left within the 
required cross-sectional figure. Farther on, in the same tunnel, 
when passing through some very soft treacherous material, it 
may be necessary to put in a full arch lining — top, sides, and 
bottom — which will be nearly circular in cross-section. For 
an illustration of this see Figs. 87 and 88. 

The width of tunnels varies as greatly as the designs. Single- 
track tunnels generally have a width of 15 to 16 feet. Occa- 
sionally they have been built 14 feet wide, and even less, and 
also up to 18 feet, especially when on curves. 24 to 26 feet is 
the most common width for double track. Many double-track 
tunnels are only 22 feet wide, and some are 28 feet wide. The 
heights are generally 19 feet for single track and 20 to 22 feet 
for double track. The variations from these figures are con- 
siderable. The lower limits depend on the cross -section of the 
rolling stock, with an indefinite allowance for clearance and ven- 
tilation. Cross-sections which coincide too closely with what is 
absolutely required for clearance are objectionable, because any 
slight settlement of the lining which would otherwise be harm- 
less would then become troublesome and even dangerous. Figs. 
87, 88, and 89 * show some typical cross-sections. 

165. Grade. A grade of at least 2% is needed for drainage. 
If the tunnel is at the summit of two grades, the tunnel grade 
should be practically level, with an allowance for drainage, the 

* Drinker's "Tunneling." 



§165. 



TUNNELS. 



195 




Fig. 87. — Hoosac Tunnel. Section through Solid Rock, 




Fig, 88. — Hoosac Tunnel. Section through Soft Ground. 



196 



RAILROAD CONSTRUCTION. 



§166. 



actual summit being perhaps in the center so as to drain both 
ways. When the tunnel forms part of a long ascending grade, 
it is advisable to reduce the grade through the timnel unless the 
tunnel is very short The additional atmospheric resistance and 
the decreased adhesion of the driver wheels on the damp rails in 
a tunnel will cause an engine to work very hard and still more 
rapidly vitiate the atmosphere until the accumulation of poison- 
ous gases becomes a source of actual danger to the engineer and 




Fir.. 89. St. Cloud Tunnel. 

fireman of the locomotive and of extreme discomfort to the 
passengers. If the nominal ruling grade of the road were 
maintained through a tunnel, the maximum resistance would be 
found in the tunnel. This would probably cause trains to stall 
there, which would be objectionable and perhaps dangerous. 

1 66. Lining. It is a characteristic of many kinds of rock 
and of all earthy material that, although they may be self- 
sustaining when first exposed to the atmosphere, they rapidly 
disintegrate and require that the top and perhaps the ?ides and 
even the bottom shall be lined to prevent caving in. In this 
country, when timber is cheap, it is occasionally framed as an 
arch and used as the jpermanent lining, but masonry is always 
to be preferred. Frequently the cross-section is made extra 



§167. 



TUNNELS. 



197 



large so that a masonry lining may subsequently be placed inside 
the wooden lining and thus postpone a large expense until the 
road is better able to pay for the work. In very soft unstable 
material, like quicksand, an arch of cut stone voussoirs may be 
necessar}^ to withstand the pressure. A good quality of brick is 
occasionally used for lining, as they are easily handled and make 
good masonry if the pressure is not excessive. Only the best 
of cement mortar should be used, economy in this feature being 
the worst of folly. Of course the excavation must include the 
outside line of the lining. Any excavation which is made out- 
side of this line (by the fall of earth or loose rock or b}- exces- 
sive blasting) must he refilled with stone well packed in. Occa- 
sionally it is necessary to fill these spaces with concrete. Of 
course it is not necessary that the lining be uniform throughout 
the tunnel. 

167. Shafts. Shafts are variously made with square, rectan- 
gular, elliptical, and circular cross-sections. The rectangular 




Fig. 90. — Connection with Shaft, Chfrch Hill Tunnel. 



cross-section, with the longer axis parallel with the tunnel, is 
most usually employed. Generally the shaft is directly over the 
center of the tunnel, but that always impHes a complicated con- 
nection between the Hnings of the tunnel and shaft, provided 



198 



KAILROAD CONSTRUCTION. 



§ 168. 



such linings are necessary. It is easier to sink a shaft near to 
one side of the tunnel and make an opening through the nearly 
vertical side of the tunnel. Such a method was employed in the 
Church Hill Tunnel, illustrated in Fig. 90.* Fig. 91 t shows 
a cross-section for a large main shaft. Many shafts have been 
built with the idea of being left open permanent h' for ventila- 
tion and have therefore been elaborately lined with masonry. 




Fig. 91. — Cross-section. Large Main Shaft. 

The general consensus of opinion now appears to be that shafts 
are worse than useless for ventilation; that the quick passage of 
a train through the tunnel is the most effective ventilator; and 
that shafts only tend to produce cross-currents and are ineffective 
to clear the air. In consequence, many of these elaborately 
lined shafts have been permanently closed, and the more recent 
practice is to close up a shaft as soon as the tunnel is completed. 
Shafts always form drainage -wells for the material they pass 
through, and sometimes to such an extent that it is a serious 
matter to dispose of the water that collects at the bottom, 
requiring the construction of large and expensive drains. 

1 68. Drains. A tunnel will almost invariably strike veins of 
water which will promptly begin to drain into the tunnel and 
not only cause considerable trouble and expense during construc- 
tion, but necessitate the provision of permanent drains for its 
perpetual disposal. These drains must frequently be so large as 



* Drinker's "Tunneling." 

t Rziha, "Lehrbuch dier Gesammten Tunnelbaukunst." 



§ 169. 



TUNNELS. 



199 



to appreciably increase the required cross-section of the tunnel. 
Generally a small open gutter on each side will suffice for this 
purpose, but in double-track tunnels a large covered drain is 
often built between the tracks. It is sometimes necessary to 
thoroughly grout the outside of the lining so that water mil not 
force its way through the masonry and perhaps injure it, but 
may freely drain down the sides and pass through openings in 
the side walls near their base into the gutters. 



CONSTRUCTION. 

169. Headings. The methods of all tunnel excavation de- 
pend on the general principle that all earthy material, except 
the softest of liquid mud and quicksand, will be self-sustaining 
over a greater or less area and for a greater or less time after 
excavation is made, and the work consists in excavating some 
material and immediately propping up the exposed surface by 
timbering and poling-boards. The excavation of the cross- 
section begins with cutting out a ''heading," which is a small 
horizontal drift whose breast is constantly kept 15 feet or more 
in advance of the full cross-sectional excavation. In solid 
self-sustaining rock, which will not decompose upon exposure 
to air, it becomes simply a matter of excavating the rock with 
the least possible expenditure of time and energy. In soft 
ground the heading must be heavily timbered, and as the heading 
is gradually enlarged the timbering must be gradually extended 
and perhaps replaced, according to some regular system, so that 
when the full cross -section has been ex- 
cavated it is supported by such timbering 
as is intended for it. The heading is 
sometimes made on the center line near 
the top; with other plans, on the center 
line near the bottom; and sometimes tw^o 
simultaneous headings are run in the two 
lower corners. Headings near the bot- 
tom serve the purpose of draining the 
material above it and facilitating the 
excavation. The simplest case of head- 
ing timbering is that shown in Fig. 92, 
in which cross-timbers are placed at in- yig 92 

tervals just under the roof, set in notches 
cut in the side walls and supporting poling-boards which sus- 




200 



RAILROAD CONSTRUCTION. 



§170. 



tain whatever pressure may come on them. Cross-timbers 
near the bottom support a flooring on which vehicles for trans- 
porting material may be run and under which the drainage 
may freely escape. As the necessity for timbering becomes 
greater, side timbers and even bottom timbers must be added, 
these timbers supporting poling -boards, and even the breast 
of the heading must be protected by boards suitably braced. 




m .^. . ...._.„. _..,,.. „..-.m^ 

Fig. 93. — Timbering for Tunnel Heading. 

as shown in Fig. 93. The supporting timbers are framed into 
collars in such a manner that added pressure only increases 
their rigidity. 

170. Enlargement. Enlargement is accomplished by remov- 
ing the poling-boards, one at a time, excavating a greater or less 
amount of material, and immediately supporting the exposed 
material with poling-boards suitably braced. (See Figs. 93 and 
94.) This work being systematically done, space is thereby 
obtained in which the framing for the full cross-section ma}^ be 
gradually introduced. The framing is constructed with a cross- 



§171. 



TUNNEI,S. 



201 



section so large that the masonr}^ lining may be constructed 
within it. 

171. Distinctive features of various methods of construction. 
There are six general systems, known as the English, German, 
Belgian, French, Austrian, and American. They are so named 




Fig. 94. 



from the origin of the methods, although their use is not con- 
fined to the countries named. Fig. 95 shows by numbers (1 to 5) 
the order of the excavation within the cross-sections. The Eng- 
lish, Austrian, and American systems are alike in excavating the 
entire cross-section before beginning the construction of the 
masonry lining. The German method leaves a solid core (5) 
until practically the whole of the lining is complete. This has 
the disadvantage of extremely cramped quarters for work, poor 
ventilation, etc. The Belgian and French methods agree in 
excavating the upper part of the section, building the arch at 
once, and supporting it temporarily until the side w^alls are 
built. The Belgian method then takes out the core (3), removes 
very short sections of the sides (4) immediately underpinning 
the arch with short sections of the side walls and thus gradually 
constructing the whole side wall. The French method digs out 
the sides (3), supporting the arch temporarily w^ith timbers and 
then replacing the timbers with masonry; the core (4) is taken 
out last. The French method has the same disadvantage as the 
German— -working in a cramped space. The Belgian and French 
systems have the disadvantage that the arch, supported tempo- 
rarily on timber, is very apt to be strained and cracked by the 
slight settlement that so frequently occurs in soft material. The 
English, Austrian, and American methods differ mainly in the 



202 



KAILROAD CONSTRUCTION. 



171. J 

.1 



design of the timbering. The English support the roof by hnes 
of very heavy longitudinal timbers which are supported at com- 
paratively wide intervals by a heavy framework occupying the 



4 i 


1 


|A 


3 


i 4 







— i 


5 1 

1 


1 


1 5 




ENGLISH 



AUSTRIAN 




GERMAN 




BELGIAN 




FRENCH 




AMERICAN 



Fig. 95. — Order of Working by the Various Systems. 

whole cross-section. The Austrian system uses such frequent 
cross-frames of timber-work that poling-boards will suffice to 
support the material between the frames. The American sys- 
tem agrees with the Austrian in using frequent cross-frames 



§ 172. TUNNELS. 203 

supporting poling-boards, but differs from it in that the "cross- 
frames" consist simpl}^ of arches of 3 to 15 wooden voussoirs, 
the voussoirs being blocks of 12''Xl2'' timber about 2 to 8 feet 
long and cut with joints normal to the arch. These arches are 
put together on a centering which is removed as soon as the arch 
is keyed up and thus immediately opens up the full cross-section, 
so that the center core (4) may be immediately dug out and the 
masonry constructed in a large open space. The American sys- 
tem has been used successfully in ver}- soft ground, but its ad- 
vantages are greater in loose rock, when it is much cheaper than 
the other methods which employ more timber. Fig. 90 and 
Plate III illustrate the use of the American system. Fig. 90 
shows the wooden arch in place. The masonry arch may be 
placed when convenient, since it is possible to lay the track and 
commence traffic as soon as the wooden arch is in place. The 
student is referred to Drinker's ^^ Tunneling" and to Rziha's 
^'Lehrbuch der Gesammten Tunnelbaukunst " for numerous 
illustrations of European methods of tunnel timbering. 

172. Ventilation during construction. Tunnels of any great 
length must be artificially ventilated during construction. If 
the excavated material is rock so that blasting is necessar}^, the 
need for ventilation becomes still more imperative. The inven- 
tion of compressed-air drills simultaneously solved two difficul- 
ties. It introduced a motive power which is unobjectionable in 
its application (as gas would be), and it also furnished at the same 
time a supply of just what is needed — pure air. If no blasting 
is done (and sometimes even when there is blasting), air must be 
supplied by direct pumping. The cooling effect of the sudden 
expansion of compressed air only reduces the otherwise objection- 
ably high temperature sometimes found in tunnels. Since pure 
air is being continually pumped in, the foul air is thereby forced 
out. 

173. Excavation for the portals. Under normal conditions 
there is always a greater or less amount of open cut preceding 
and following a tunnel. Since all tunnel methods depend (to 
some slight degree at least) on the capacity of the exposed ma- 
terial to act as an arch, there is implied a considerable thickness 
of material above the tunnel. This thickness is reduced to 
nearly zero over the tunnel portals and therefore requires special 
treatment, particularl}^ when the material is very soft. Fig. 96 * 

* Rziha, "Lehrbuch der Gesammten Tunnelbaukunst." 



204 



RAILROAD CONSTRUCTION. 



§ 174. 



illustrates one method of breaking into the ground at a portal. 
The loose stones are piled on the framing to give stability to the 
framing by their weight and also to retain the earth on the 




Fig. 96. — Timberixg for Tunnel Portal. 



slope above. Another method is to sink a temporary shaft to 
the tunnel near the portal; immediately enlarge to the full size 
and build the masonry lining; then work back to the portah 
This method is more costly, but is preferable in very treacherous 
ground, it being less liable to cause landslides of the surface 
material. 

174. Tunnels vs. open cuts. In cases in which an open cut 
rather than a tunnel is a possibility the ultimate consideration 
is generally that of first cost combined wdth other financial con- 



/-V^/-/ ^/-^^ 



PLATE III. 




(ro lace pafifg 205.) 



Elevation of Portal. 
Phcenixviw-e Tdnnel, p. S. V. R. R. 



PLATE III. 




Longitudinal Section of PortaIj. 



§ 175. 



TUNNELS. 



205 



siderations and annual maintenance charges directly or indirectly 
connected with it. Even when an open cut may be constructed 
at the same cost as a tunnel (or perhaps a little cheaper) the 
tunnel may be preferable under the following conditions: 

1. When the soil indicates that the open cut would be liable 
to landslides. 

2. When the open cut would be subject to excessive snow- 
drifts or avalanches. 

3. When land is especially costly or it is desired to nm under 
existing costly or valuable buildings or monuments. When run- 
ning through cities, tunnels are sometimes constructed as open 
cuts and then arched over. 

These cases apply to tunnels vs. open cuts when the align- 
ment is fixed by other considerations than the mere topography. 
The broader question of excavating tunnels to avoid excessive 
grades or to save distance or curvature, and similar problems, 
are hardly susceptible of general analysis except as questions of 
railway economics and must be treated individually. 

175. Cost of tunneling. The cost of any construction which 
involves such uncertainties as tunneling is very variable. It 
depends on the material encountered, the amount and kind of 
timbering required, on the size of the cross-section, on the price 
of labor, and especially on the reconstruction that may be neces- 
sary on account of mishaps. 

Headings generally cost $4 to $5 per cubic yard for excava^ 
tion, while the remainder of the cross-section in the same tunnel 
may cost about half as much. The average cost of a large 
number of tunnels in this country may be seen from the follow- 
ing table:* 





Cost per cubic yard. 


Cost per 
lineal foot. 




Excavation. 


Masonry. 




Material. 


Single. 






Single. 


Double. 


Single. 


Double. 


Double. 


Hard rock 

Loose rock 

Soft ground.. . . 


$5.89 
3.12 
3.62 


$5 . 45 
3.48 
4.64 


$12.00 

9.07 

15.00 


$8.25 
10.41 
10.50 


$69 . 76 

80.61 

135.31 


$142.82 
119.26 
174.42 



* Figures derived from Drinker's "Tunneling." 



206 RAILROAD CONSTRUCTION. § 175. 

A considerable variation from these figures may be found in 
individual cases, due sometimes to unusual skill (or the lack of 
it) in prosecuting the work, but the figures will generally be 
sufficiently accurate for preliminary estimates or for the com- 
parison of two proposed routes. 



CHAPTER VI. 
CULVERTS AND MINOR BRIDGES. 

176. Definition and object. Although a variable percentage 
of the rain falling on any section of country soaks into the 
ground and does not immediately reappear, yet a very large 
percentage flows over the surface, always seeking and following 
the lowest channels. The roadbed of a railroad is constantly 
intersecting these channels, which frequently are normally dry. 
In order to prevent injurj^ to railroad embankments by the im- 
pounding of such rainfall, it is necessary to construct waterways 
through the embankment through w^hich such rainflow may 
freely pass. Such waterways, called culverts, are also appli- 
cable for the bridging of very small although perennial streams, 
and therefore in this work the term culvert will be applied to 
all water-channels passing through a railroad embankment 
which are not of sufficient magnitude to require a special struc- 
tural design, such as is necessary for a large masonry arch or a 
truss bridge. 

177. Elements of the design. A well-designed culvert must 
afford such free passage to the water that it will not ^'back up'' 
over the adjoining land nor cause any injury to the embankment 
or culvert. The ability of the culvert to discharge freely all the 
water that comes to it evidently depends chiefly on the area of 
the waterway, but also on the form, length, slope, and materials 
of construction of the culvert and the nature of the approach 
and outfall. When the embankment is very low and the amount 
of water to be discharged very great, it sometimes becomes 
necessary to allow the water to discharge ^' under a head," i e., 
with the surface of the water above the top of the culvert. 
Safety then requires a much stronger construction than would 
otherwise be necessary to avoid injury to the culvert or embank- 
ment by washing. The necessity for such construction should 
be avoided if possible. 

207 



208 RAILROAD CONSTRtJCTION. § 178. 



AREA OF THE WATERWAY. 

178. Elements involved. The determination of the required 
area of the waterway involves such a multiplicity of indeter- 
minate elements that any close determination of its value from 
purely theoretical considerations is a practical impossibility. 
The principal elements involved are: 

a. Rainfall. The real test of the culvert is its capacity to 
discharge without injury the flow resulting from the extraordi- 
nary rainfalls and "cloud bursts" that may occur once jn many 
years. Therefore, while a knowledge of the average annual 
rainfall is of very little value, a record of the maximum rainfall 
during heavy storms for a long term of years may give a relative 
idea of the maximum demand on the culvert. 

b. Area of watershed. This signifies the total area of country 
draining into the channel considered. When the drainage area 
is very small it is sometimes included within the area surveyed 
by the preliminary survey. When larger it is frequently possi- 
ble to obtain its area from other maps with a percentage of 
accuracy sufficient for the purpose. Sometimes a special survey 
for the purpose is considered justifiable. 

c. Character of soil and vegetation. This has a large in- 
fluence on the rapidity with which the rainflow from a given 
area will reach the culvert. If the soil is hard and impermeable 
and the vegetation scant, a heavy rain will run off suddenly, 
taxing the capacity of the culvert for a short time, while a 
spongy soil and dense vegetation will retard the flow, making it 
more nearly uniform and the maximum flow at any one time 
much less. 

d. Shape and slope of watershed. If the watershed is very 
long and narrow (other things being equal), the water from the 
remoter parts will require so much longer time to reach the 
culvert that the flow will be comparatively uniform, especially 
when the slope of the whole watershed is very low. When the 
slope of the remoter portions is quite steep it may result in the 
nearly simultaneous arrival of a storm-flow from all parts of the 
watershed, thus taxing the capacity of the culvert. 

e. Effect of design of culvert. The principles of hydraulics 
show that the slope of the culvert, its length, the form of the 
cross-section, the nature of the surface, and the form of the 



§ 179. CULVERTS AND MINOR BRIDGES. 209 

approach and discharge all have a considerable influence on the 
area of cross-section required to discharge a given volume of 
water in a given time, but unfortunately the combined hy- 
draulic effect of these various details is still a very uncertain 
quantity. 

179. Methods of computation of area. There are three pos- 
sible methods of computation. 

(a) Theoretical. As shown above it is a practical impossi- 
bility to estimate correctly the combined effect of the great mul- 
tiplicity of elements which influence the final result. The nearest 
approach to it is to estimate by the use of empirical formulae 
the amount of water which will be presented at the upper end 
of the culvert in a given time and then to compute, from the 
principles of hydraulics, the rate of flow through a culvert of 
given construction, but (as shown in § 178, e) such methods are 
still very unreliable, owing to lack of experimental knowledge. 
This method has apparently greater scientific accuracy than 
other methods, but a little study will show that the elements 
of uncertainty are as great and the final result no more reliable. 
The method is most reliable for streams of uniform flow, but 
it is under these conditions that method (c) is most useful. The 
theoretical method will not therefore be considered further. 

(b) Empirical. As illustrated in § 180, some formulae make 
the area of waterway a function of the drainage area, the for- 
mula bemg affected by a coefficient the value of which is esti- 
mated between limits according to the judgment Assuming 
that the formulae are sound, their use only narrows the limits of 
error, the final determination depending on experience and 
judgment. 

(c) From observation. This method, considered by far the 
best for permanent work, consists m observing the high-water 
marks on contracted channel-openings which are on the same 
stream and as near as possible to the proposed culvert. If the 
country is new and there are no such openings, the wisest plan 
is to bridge the opening by a temporary structure in wood which 
has an ample waterway (see § 126, b, 4) and carefully observe 
all high-water marks on that openmg during the 6 to 10 years 
which is ordinarily the minimum life of such a structure. As 
shown later, such observations may be utihzed for a close com- 
putation of the required waterway. Method (6) may be utilized 
for an approximate calculation for the required area for the tern- 



210 RAILROAD CONSTRUCTION. § 180. 

porary structure, using a value which is intentionally excessive, 
so that a permanent structure of sufficient capacity may subse- 
quently be constructed within the temporary structure. 

1 80. Empirical formulae. Two of the best known empirical 
formulae for area of the waterway are the following: 

(a) Myer's formula: 

Area of waterway in square feet = C X \/drainage area in acres, 
where C is a coefficient varying from 1 for flat country to 4 for 
mountainous country and rocky ground. As an illustration, if 
the drainage area is 100 acres, the waterway area should be from 
10 to 40 square feet, according to the value of the coefficient 
chosen. It should be noted that this formula does not regard 
the great variations in rainfall in various parts of the world nor 
the design of the culvert, and also that the final result depends 
largely on the choice of the coefficient. 

(b) Talbot's formula: 

Area of waterway in square feet = CX>y( drainage area in acres) ^. 
** For steep and rocky ground C varies from f to 1. For rolling 
agricultural country subject to floods at times of melting snow, 
and with the length of the valley three or four times its width, C 
is about i ; and if the stream is longer in proportion to the area, 
decrease C. In districts not affected by accumulated snow, and 
where the length of the valley is several times the width, I or \, 
or even less, may be used. C should be increased for steep side 
slopes, especially if the upper part of the valley has a much 
greater fall than the channel at the culvert." * As an illus- 
tration, if the drainage area is 100 acres the area of waterway 
should be (7X31.6. The area should then vary from 5 to 31 
square feet, according to the character of the country. Like 
the previous estimate, the result depends on the choice of a 
coefficient and disregards local variations in rainfall, except as 
they may be arbitrarily allowed for in choosing the coeffi- 
cient. 

181. Value of empirical formulae. The fact that these for- 
mulae, as well as many others of similar nature that have been 
suggested, depend so largely upon the choice of the coefficient 
shows that they are valuable '^ more as a guide to the judgment 
than as a working rule," as Prof. Talbot explicitly declares in 

* Prof. A. N. Talbot, "Selected Papers of the Civil Engineers' Club of 
the Univ. of Illinois." 



§ 182. CULVERTS AND MINOR BRIDGES. 211 

commenting on his own formula. In short, they are chiefly valu- 
able in indicating a probable maximum and minimum between 
which the true result probably lies. 

182. Results based on Observation. As already indicated in 
§ 179, observation of the stream in question gives the most 
reliable results. If the country is new and no records of the 
flow of the stream during heavy storms has been taken, even 
the life of a temporary wooden structure may not be long enough 
to include one of the unusually severe storms which must be 
allowed for, but there will usually be some high-water mark 
which will indicate how much opening will be required. The 
following quotation illustrates this: '^A tidal estuary may gen- 
erally be safely narrowed considerably from the extreme water 
lines if stone revetments are used to protect the bank from 
wash. Above the true estuary, Avhere the stream cuts through 
the marsh, we generally find nearly vertical banks, and we are 
safe if the faces of abutments are placed even with the banks. 
In level sections of the country, where the current is sluggish, 
it is usually safe to encroach somewhat on the general width 
of the stream, but in rapid streams among the hills the width 
that the stream has cut for itself through the soil should not be 
lessened, and in ravines carrying mountain torrents the open- 
ings must be left very much larger than the ordinary appear- 
ance of the banks of the stream would seem to make neces- 
sary." * 

As an illustration of an observation of a storm-flow through 
a temporary trestle, the following is quoted : '^ Having the flood 
height and velocity, it is an easy matter to determine the vol- 
ume of water to be taken care of. I have one ten-bent pile 
trestle 135 feet long and 24 feet high over a spring branch that, 
ordinarily runs about six cubic inches per second. Last sum- 
mer during one of our heav}^ rainstorms (four inches in less 
than three hours) I visited this place and found by float obser- 
vations the surface velocity at the highest stage to be 1.9 feet 
per second. I made a high- water mark, and after the flood- 
water receded found the width of stream to be 12 feet and an 
average depth of 2f feet. This, with a surface velocity of 1.9 
feet per second, would give approximately a discharge of 50 



* J. P. Snow, Boston & Maine Railway. From Report to Association of 
Railway Superintendents of Bridges and Buildings. 1897. 



212 RAILROAD COXSTRUCTIOX. § 183. 

cubic feet, or 375 gallons, per second. Having this information 
it is easy to determine size of opening required.' ' * 

183. Degree of accuracy required. The advantages result- 
ing from the use of standard designs for culverts (as well as 
other structures) haxe led to the adoption of a comparatively 
small number of designs. The practical use made of a compu- 
tation of required waterway area is to determine which one of 
several standard designs will most nearly fulfill the require- 
ments. For example, if a 24-inch iron pipe, having an area of 
3.14 square feet, is considered to be a little small, the next size 
(30-inch) would be adopted; but a 30-inch pipe has an area of 
4.92 square feet, which is 56% larger. A similar result, except 
that the percentage of difference might not be quite so marked, 
will be found by comparing the areas of consecutive standard 
designs for stone box culverts. 

The advisability of designing a culvert to withstand any 
storm-flow that may ever occur is considered doubtful. Several 
years ago a record-breaking storm in New England carried 
away a very large number of bridges, etc., hitherto supposed 
to be safe. It was not afterward considered that the design of 
those bridges was faulty, because the extra cost of constructing 
bridges capable of withstanding such a flood, added to interest 
for a long period of years, would be enormously greater than the 
cost of repairing the damages of such a storm once or twice in 
a century. Of course the element of danger has some weight, 
but not enough to justify a great additional expenditure, for 
common prudence would prompt unusual precautions during 
or immediately after such an extraordinary storm. 

PIPE CULVERTS. 

184. Advantages. Pipe culverts, made of cast iron or earthen- 
ware, are very durable, readily constructed, moderately cheap, 
will pass a larger volume of water in proportion to the area than 
many other designs on account of the smoothness of the sur- 
face, and (when using iron pipe) may be used very close to 
the track when a low opening of large capacity is required. 
Another advantage lies in the ease with which they may be 
inserted through a somewhat larger opening that has been 

♦ A. J. Kelley, Kansas City Belt Railway. From Report to Association 
of Railway Superintendents of Bridges and Buildings. 1897. 



§ 185. CULVERTS AND MINOR BRIDGES. 2l3 

temporarily lined with wood, without disturbing the roadbed 
or track. 

185. Construction. Permanency requires that the founda- 
tion shall be firm and secure against being w^ashed out. To 
accomplish this, the soil of the trench should be hollowed out to 
fit the lower half of the pipe, making suitable recesses for the 
bells. In very soft treacherous soil a foundation-block of con- 
crete is sometimes placed under each joint, or even throughout 
the whole length. When pipes are laid through a slightly 
larger timber culvert great care should be taken that the pipes 
are properly supported, so that there will be no settling nor 
development of unusual strains w^hen the timber finally decays 
and gives way. To prevent the w^ashing away of material 
around the pipe the ends should be protected by a bulkhead. 
This is best constructed of masonrj^ (see Fig. 97), although wood 
is sometimes used for cheap and minor constructions. The joints 
should be calked, especially when the culvert is liable to run 
full or w^hen the outflow is impeded and the culvert is liable to 
be partly or wholly filled during freezing weather. The cost of 
a calking of clay or even hydraulic cement is insignificant com- 
pared with the value of the additional safety afforded. When 
the grade of the pipe is perfectly uniform, a very low rate of 
grade will suffice to drain a pipe culvert, but since some uneven- 
ness of grade is inevitable through uneven settlement or im- 
perfect construction, a grade of 1 in 20 should preferably be 
required, although much less is often used. The length of a 
pipe culvert is approximately determined as follows: 

Length = 2.^ (depth of embankment) + (width of roadbed), 

in which s is the slope ratio (horizontal to vertical) of the banks. 
In practice an even number of lengths will be used which will 
most nearly agree with this formula. 

186. Iron-pipe culverts. Simple cast-iron pipes are used in 
sizes from 12" to 48'' diameter. These are usually made in 
lengths of 12 feet with a few lengths of 6 feet, so that any required 
length may be more nearly obtained. The lightest pipes made 
are sufficiently strong for the purpose, and even those w^hich 
would be rejected because of incapacity to withstand pressure 
may be utilized for this work. In Fig. 97 are shown the stand- 
ard plans used on the C. C. C. & St. L. Ry., which may be con- 
sidered as typical plans. 




-aJ 



0,2- 



— 9;2- 



Fjg- 97. — Standard Cast-iron 
Pipe Culvert. C. C. C. & 
St. L. Ry. (May 1893.) 



214 



§187. 



CULVERTS AND MINOR BRIDGES. 



215 



Pipes formed of cast-iron segments have been used up to 12 ' 
feet diameter. The shell is then made comparatively thin, but 
is stiffened by ribs and flanges on the outside. The segments 
break joints and are bolted together through the flanges. The 
joints are made tight by the use of a tarred rope, together with 
neat cement. 

187. Tile-pipe culverts. The pipes used for this purpose 
vary from 12" to 24'' in diameter. When a larger capacity is 
required two or more pipes may be laid side by side, but in 
such a case another design might be preferable. It is frequently 
specified that "double-strength" or "extra-heavy^ pipe shall 
be used, evidently with the idea that the stresses on a culvert- 
pipe are greater than on a sewer-pipe. But it has been con- 
clusively demonstrated that, no matter how deep the embank- 
ment, the pressure cannot exceed a somewhat unceitain maxi- 
mum, also that the greatest danger consists in placing the pipe 
so near the ties that shocks may be directly transferred to the 
pipe -without the cushioning effect of the earth and ballast. 
When the pipes are well bedded in clear earth and there is a 




UP-STR-EAr/.„END. DOWN-STREAM EM D. DOWN-STREAM EN.D. THREE PIPE&* 

Fig. 98. — Standard Vitrified-pipe Culvert. Plant System. (1891.) 

sufficient depth of earth over them to avoid direct impact (at 
least three feet) the ordinary sewer-pipe will be sufficiently 
strong. ''Double-strength" pipe is frequently less perfectly 
burned, and the supposed extra strength is not therefore ob- 



216 



RAILROAD COXSTRUCTIOX. 



§18S. 



'tained. In Fig. 98 are shown the standard plans for vitrified- 
pipe culverts as used on the ^' Plant system." Tile pipe is much 
cheaper than iron pipe, but is made in much shorter lengths and 
requires much more work in laying and especially to obtain a 
uniform grade. 

BOX CULVERTS. 

1 88. Wooden box culverts. This form serves the purpose 
of a cheap temporan^ construction which allows the use of a 
ballasted roadbed. As in all temporary constructions, the area 
should be made considerably larger than the calculated area 
§§ 179-182), not only for safety but also in order that, if the 
smaller area is demonstrated to be sufficiently large, the per- 
manent construction (probably pipe) may be placed inside with- 
out disturbing the embankment. All designs agree in using 
heavy timbers (12"Xl2", 10''Xl2'', or 8"Xl2'0 for the side 
walls, cross-timbers for the roof, every fifth or sixth timber 
being notched down so as to take up the thrust of the side walls, 
and planks for the flooring. Fig. 99 shows some of the standard 
designs as us^d by the C, M & St. P. Ry. 



C10TE:-F0R 6 COVERING^ EVERY SIXTH STICK 8 THICK. 





^^^^^^^^^^^^^^^^^^ 



Fig. 99. — Standard Timber Box Culvert. C, M. 6c St. P. Rt. 
(Feb. 1889.) 

189. Stone box culverts. In localities where a good quahty 
of stone is cheap, stone box culverts are the cheapest form of 
permanent construction for culverts of medium capacity, but 
their use is decreasing owing to the frequent difficulty in obtain- 
ing really suitable stone within a reasonable distance of the 
culvert. The clear span of the cover-stones varies from 2 to 4 
feet. The required thickness of the cover-stones is sometimes 



§ 189. 



CULVERTS AND MINOR BRIDGES. 



217 



calculated by the theory of transverse strains on the basis of 
certain assumptions of loading — as a function of the height of 
the embankment and the unit strength of the stone used. Such 
a method is simply another illustration of a class of calculations 
which look very precise and beautiful, but which are worse than 
useless (because misleading) on account of the hopeless uncer- 




PLAN 

Fig. 100.— Stand .A.RD Single Stone Culvert (3'X4'). N. & W. R.R. 

(1890.) 

tainty as to the true value of certain quantities which must be 
used in the computations In the first place the true value of 
the unit tensile strength of stone is such an uncertain and variable 



218 



RAILROAD CONSTRUCTION. 



189. 



quantity that calculations based on any assumed value for it are 
of small reliability. In the second place the weight of the prism 
of earth lying directly above the stone, plus an allowance for live 
load, is by no means a measure of the load on the stone nor of 
the forces that tend to fracture it. All earthwork will tend to 




PLAN 



Fig. 100a. — Standard Double Stone Culvert (3'X40. N. <fe W. R. R. 

(1890.) 



form an arch above any cavity and thus relieve an uncertain 
and probably variable proportion of the pressure that might 
otherwise exist. The higher the embankment the less the pro- 



190. 



CULVERTS AND MINOR BRIDGES. 



219 



portionate loading, until at some uncertain height an increase 
in height will not increase the load on the cover-stones. The 
effect of frost is likewise large, but uncertain and not computable. 
The usual practice is therefore to make the thickness such as 
experience has shown to be safe with a good quality of stone, 
i.e., about 10 or 12 inches for 2 feet span and up to 16 or 18 
inches for 4 feet span. The side walls should be carried down 
deep enough to prevent their being undermined by scour or 
heaved by frost. The use of cement mortar is also an important 
feature of first-class work, especially when there is a rapid scour- 
ing current or a liability that the culvert will run under a head. 
In Figs. 100 and 100a are shown standard plans for single and 
doublejstone box culverts as used on the Norfolk^and Western R.R. 
190. Old-rail culverts. It sometimes happens (although very 
rarely) that it is necessary to bring the grade line within 3 or 4 
feet of the bottom of a stream and yet aUow an area of 10 or 12 
square feet. A single large pipe of sufficient area could not be 
used in this case. The use of several smaller pipes side by side 
would be both expensive and inefficient. For similar reasons 
neither wooden nor stone box culverts could be used. In such 
cases, as well as in many others where the head-room is not so 
limited, the plan illustrated in Fig. 101 is a very satisfactory 



i^^^^^Si 




•""^ BROKEN STONE BALLAST £1 ^*^^ 

TrrTTTTTTr T,TTTTTT.,TTTt TTTTTTTTTTTTTTr:7?1 




Fio. 101.— Standard Old-rail Culvert. N. & W. R.R. (1895.) 

solution of the problem. The old rails, having a length of 8 or 
9 feet, are laid close together across a 6-foot opening. Some- 
times the rails are held together by long bolts passing through 



220 RAILROAD CONSTRUCTION. § 190. 

the webs of the rails. In the plan shown the rails are confined 
by low end walls on each abutment. This plan requires only 
15 inches between the base of the rail and the top of the culvert 
channel. It also gives a continuous ballasted roadbed. 

iQoa. Reinforced Concrete Culverts. The development of 
reinforced concrete as a structural material is illustrated in its 
extensive adoption for arches and also for culverts. One of the 
special types which has been adopted is that of a box culvert 
which has a concrete bottom. Since this bottom can be made 
so that it will withstand an upward transverse stress, it furnishes 
a broad foundation for the whole culvert, and thus entirely 
eliminates the necessity for extensive footing to the side walls of 
the culvert, such as are necessary in soft ground with an ordinary 
stone culvert. Another advantage is that the inside of the cul- 
vert may be made perfectly smooth and thus offer less resistance 
to the passage of water through it. As may be noticed from 
Fig. 101a, such a culvert is provided with flaring head walls, and 
sunken end walls, so that the water may not scour underneath 
the culvert, and other features common to other types. No 
attempt will here be made to discuss the design of reinforced 
concrete, except to say that all four sides of such a box culvert 
are designed to withstand a computed bursting pressure which 
tends to crush the flat sides inward. In Fig. 101a is shown one 
illustration of the many types of culverts which have been de- 
signed of reinforced concrete. 



ARCH CULVERTS. 

191. Influence of design on flow. The variations in the design 
of arch culverts have a very marked influence on the cost and 
efficiency. To combine the least cost with the greatest effi- 
cienc}^ due weight should be given to the following elements: 
(a) amount of masonr}-. (6) the simplicity of the constructive 
work, (c) the design of the wing walls, (d) the design of the 
junction of the wing walls with the barrel and faces of the arch, 
and (e) the safety and permanency of the construction. These 
elements are more or less antagonistic to each other, and the 
defects of most designs are due to a lack of proper proportion 
in the design of these opposing interests. The simplest con- 
struction (satisfying elements b and e) is the straight barrel arch 



STANDARD ARCH CULVERT 

8FKTSPAN 

XJORFOLK & WESTERN R.R. 

(1891) 



I'j^h^ 




(To face page 22 h) 



§192. 



CULVERTS AND MINOR BRIDGES. 



221 



-J 



i 



-0 9- 




— '---f-^-^-f 1--— H- 

-h-r-t-fi'it- 
-i— ^— f-t^-f- 




-oi-s-^ 




i>- 



^^^^^^H 



-Sl- 




-z 
g 
I- 
o 



CO 
CO 

O 
or 
o 




222 



RAILROAD CONSTRUCTION. 



§ 191. 



between two parallel vertical head walls, as sketched in Fig. 
102, a. From a hydraulic standpoint the design is poor, as the 
water eddies around the corners, causing a great resistance 
which decreases the flow. Fig. 102, h, shows a much better de- 




FiG. 102. — Types of Culverts. 

sign in many respects, but much depends on the details of the 
design as indicated in elements (b) and {d). As a general thing 
a good hydraulic design requires complicated and expensive 
masonry construction, i.e., elements (6) and {d) are opposed. 
Design 102, c, is sometimes inapplicable because the water is 
liable to work in behind the masonry during floods and perhaps 
cause scour. This design uses less masonry than (a) or {h). 

192. Example of arch culvert design. In Plate IV is shown 
the design for an 8-foot arch culvert according to the standard 
of the Norfolk and Western R. R. Note that the plan uses the 
flaring wing walls (Fig. 102, h) on the up-stream side (thus 
protecting the abutments from scour) and straight wing walla 
(similar to Fig. 102, c) on the down-stream end. This econo- 
mizes masonry and also simplifies the constructive work. Note 
also the simplicity of the junction of the wing walls with the 
barrel of the arch, there being no re-entrant angles below the 
springing line of the arch. The design here shown is but one 
of a set of designs for arches varying in span from 6' to 30'. 



MINOR OPENINGS. 



193. Cattle-guards, (a) Pit guards. Cattle-guards will be 
considered under the head of minor openings, since the old- 
fashioned plan of pit guards, which are even now defended and 



§ 193. 



CULVEKTS AND MINOR BRIDGES. 



223 



preferred by some railroad men, requires a break in the con- 
tinuity of the roadbed. A pit about three feet deep, five feet 




Fig. 103. — Cattle-guard with Wooden Slats. 



long, and as wide as the width of the roadbed, is walled up with 
stone (sometimes with wood), and the rails are supported on 
heavy timbers laid longitudinally with the rails. The break in 
the continuity of the roadbed produces a disturbance in the 
elastic wave running through the rails, the effect of which is 
noticeable at high velocities. The greatest objection, however, 
lies in the dangerous consequences of a derailment or a failure 
of the timbers owing to unobserved decay or destruction by 
fire — caused perhaps by sparks and cinders from passing loco- 
motives. The very insignificance of the structure often leads 
to careless inspection. But if a single pair of wheels gets off the 
rails and drops into the pit, a costly wreck is inevitable. 

(b) Surface cattle-guards. These are fastened on top of the 
ties; the continuity of the roadbed is absolutely unbroken and 
thus is avoided much of the danger of a bad wreck owing to a 
possible derailment. The device consists essentially of overlay- 
ing the ties (both inside and outside the rails) with a surface on 



KAILKOAD CONSTRUCTION. 



§193. 



which cattle will not walk. The multitudinous designs for such 
a surface are variously effective in this respect. An objection, 




CENTER SECTION 



Fio. 104. — Sheffield Cattle-ouarb. . 



.^^^ 




Fio. 105. — Climax Cattle-guard (tile). 



wh'icm is often urged indiscriminately against all such designs, is 
the liability that a brake-chain which may happen to be drag- 
ging may catch in the rough bars which are used. The bars 



§ 195. CULVERTS AND MINOR BRIDGES. 225 

are sometimes "home-made,'^ of wood, as shown in Fig. 103. 
Steel guards may be made as shown in Fig. 104. The general 
construction is the same as for the wooden bars. The metal 
bars have far greater durability, and it is claimed that they are 
more effective in discouraging cattle from attempting to cross. 

194. Cattle-passes. Frequently when a railroad crosses a 
farm on an embankment, cutting the farm into two parts, the 
railroad company is obliged to agree to make a passageway 
through the embankment sufficient for the passage of cattle and 
perhaps even farm-wagons. If the embankment is high enough 
so that a stone arch is practicable, the initial cost is the only 
great objection to such a construction; but if an open wooden 
structure is necessary, all the objections against the old-fashioned 
cattle-guards apply with equal force here. The avoidance of a 
grade crossing which would otherwise be necessary is one of the 
great compensations for the expense of the construction and 
maintenance of these structures. The construction is some- 
times made by placing two pile trestle bents about 6 to 8 feet 
apart, supporting the rails by stringers in the usual way, the 
special feature of this construction being that the embankments 
are filled in behind the trestle bents, and the thrust of the em- 
bankments is mutually taken up through the stringers, which 
are notched at the ends or otherwise constructed so that they 
may take up such a thrust. The designs for old-rail culverts 
and arch culverts are also utilized for cattle-passes when suitable 
and convenient, as well as the designs illustrated in the following 
section. 

195. Standard stringer and I-beam bridges. The advantages 
of standard designs apply even to the covering of short spans 
with wooden stringers or with I beams — especially since the 
methods do not require much vertical space between the rails 
and the upper side of the clear opening, a feature which is often 
of prime importance. These designs are chiefly used for cul- 
verts or cattle-passes and for crossing over highways — providing 
such a narrow opening would be tolerated. The plans all imply 
stone abutments, or at least abutments of sufficient stability to 
withstand all thrust of the embankments. Some of the designs 
are illustrated in Plate Y. The preparation of these standard 
designs should be attacked by the same general methods as 
already illustrated in § 156. When computing the required 



226 RAILROAD CONSTRUCTION. § 195. 

transverse strength, due allowance should be made for lateral 
bracing, which should be amply provided for. Note particu- 
larly the methods of bracing illustrated in Plate V. The designs 
calling for iron (or steel) stringers may be classed as permanent 
constructions, which are cheap, safe, easily inspected and main- 
tained, and therefore a desirable method of construction. 




(i'o face page 226.) 



CHAPTER VII. 

BALLAST. 

196. Purpose and requirements. "The object of the ballast 
is to transfer the applied load over a large surface ; to hold the 
timber work in place horizontally; to carry off the rain-water 
from the superstructure and to prevent freezing up in winter; 
to afford means of keeping the ties truly up to the grade line; 
and to give elasticity to the roadbed.'' This extremely con- 
densed statement is a description of an ideally perfect ballast. 
The value of any given kind of ballast is proportional to the 
extent to which it fulfills these requirements. The ideally 
perfect ballast is not necessarily the most economical ballast 
for all roads. Light traffic generally justifies something cheaper, 
but a very common error is to use a very cheap ballast when a 
small additional expenditure would procure a much better 
ballast, which would be much more economical in the long run. 

197. Materials. The materials most commonly employed are 
gravel and broken stone. In many sections of the country 
other materials which more or less perfectly fulfill the require- 
ments as given above, are used. The various materials includ- 
ing some of these special types have been defined by the American 
Railway Engineering and Maintenance of Way Association as 
follows: 

Definitions. 

Ballast. Selected material placed on the roadbed for the 
purpose of holding the track in line and surface. 

Broken or crushed stone. Stone broken by artificial means 
into small fragments of specified sizes. 

Burnt clay. A clay or gumbo which has been burned into 
material for ballast. 

227 



228 RAILROAD CONSTRUCTION. § 197. 

Chats. Tailings from mills in which zinc and lead ores are 
separated from the rocks in which they occur. 

Chert. An impure flint or hornstone occurring in beds. 

Cinders. The residue from the fuel used in locomotives and 
other furnaces. 

Gravel. Small worn fragments of rock, coarser than sand, 
occurring in natural deposits. 

Gumbo. A term commonly used for a peculiarly tenacious 
clay, containing no sand. 

Sand. Any hard, granular, comminuted rock material, finer 
than gravel and coarser than dust. 

Slag. The waste product, in a more or less vitrified form, of 
furnaces for reduction of ore. Usually the product of a blast- 
furnace. 

There is still another classification which may or may not be 
considered as ballast. It is perhaps hardly correct to speak 
of the natural soils as ballast, yet many miles of cheap rail- 
ways are '^ ballasted'' with the natural soil, which is then called 
Mud ballast. 

Broken or crushed stone. Rock ballast is generally specified 
to be that which may all be passed through a 1^ inch (or 2 inch) 
ring, but which cannot pass through a |-inch mesh. It is most 
easily handled with forks. This method also has the advantage 
that when it is being rehandled the fine chips which would 
interefere with effectual drainage will be screened out. Rock 
ballast is more expensive in first cost and is also more trouble- 
some to handle, but in heavy traffic especially, the track will be 
kept in better surface and will require less work for maintenance 
after the ties have become thoroughly bedded. 

Burnt clay. Tills material has been used in many sections of 
the country where broken stone or gravel are unobtainable 
except at a prohibitive cost, and where a suitable quality of 
clay is readily obtained. This clay should be of '^ gumbo'* 
variety and contain no gravel. It is sometimes burnt in a 
kiln, or it is sometimes burnt by piling the clay in long heaps 
over a mass of fuel, the pile being formed in such a way that 
a temporary but effectual kiln is made. It is necessary that 
a clear, clean fuel shall be used and that the fii'ing shall be 
done by a man who is experienced in maintaining such a fire 
until the burning is completed. Such ballast may be burned 
very hard and it will last from four to six years. The cost of 



§ 197. BALLAST. 229 

burning varies from 30 to 60 cents per cubic yard, according 
to the circumstances. 

Chats. This is a form of ballast which is peculiar to South- 
western Missouri and Southeastern Kansas. When this mate- 
rial was first used it was obtained from the refuse piles of the 
mills which treated the zinc and lead ores mined in those regions. 
With the processes then employed the material was obtained 
in lumps as large as broken stone, and they were considered to 
be as valuable as broken stone for ballast. Improvements in 
the processes of treating the ores have resulted in making this 
by-product very much smaller grained and of less value as bal- 
last, although it is still considered a desirable form of ballast 
where it may readily be obtained. It should be noted that it 
is classed with gravel and cinders in the forms of cross-section 
shown later. 

Chert. This is a form of flint or hornstone which occurs in 
nodules of a size that is suitable for ballast, and is a very good 
type of ballast wherever it is found, but its occurrence is com- 
paratively infrequent. It is classed with cemented gravel in 
the design of cross-sections of ballast. 

Cinders. This is one of the most universal forms of ballast, 
since it is a by-product of every road which uses coal as fuel. 
The advantages consist in the fairly good drainage, the ease of 
handling and the cheapness — after the road is in operation. 
One of the greatest disadvantages is the fact that the cinders are 
readily reduced to dust, which in dry weather becomes very 
objectionable. Cinders are usually considered preferable to 
gravel in yards. 

Gravel. This is one of the most common forms of good 
ballast. There are comparatively few railroads which cannot 
find, at some place along their line, a gravel pit which will 
afford a suitable supply oi gravel for ballast. Sometimes it is 
unnecessary to screen it, but usually it is better to screen the 
gravel over a screen having a J-inch mesh so as to screen 
out all the dirt and the finer stones. 

Sand. Railroads which run along the coast are frequently 
ballasted merely with the sand obtained in the immediate 
neighborhood. One great advantage lies in the almost perfect 
drainage which is obtained. 

Slag. When slag is readily obtainable it furnishes an ex- 
cellent ballast which is free from dust and perfect in drainage 



230 RAILROAD CONSTRUCTION. § 197. 

qualities. Slag is classified with crushed rock in the cross- 
sections shown below, but it should be noted that this only 
applies to the best qualities of slag, since its quality is quite 
variable. 

Mud ballast. When the natural soil is gravelly so that rain 
will drain through it quickly, it will make a fair roadbed for 
light traffic, but for heavy traffic, and for the greater part of 
the length of most roads, the natural soil is a very poor material 
/for ballast ; for, no matter how suitable the soil might be along 
limited sections of the road, it would practically never happen 
that the soil would be uniformly good throughout the whole 
length of the road. Considering that a heavy rain will in one 
day spoil the results of weeks of patient *' surfacing" with mud 
ballast, it is seldom economical to use "mud" if there is a 
gravel-bed or other source of ballast anywhere on the line of 
the road. 

198. Cross-sections. The required depth of the cross-section 
to th© sub-soil depends largely on the weight of the rolling 
stock which is to pass over the track. A careful examination 
of a roadbed to determine the changes which take place under 
th« ties and also an examination of the track and ties during 
the passage of a heavy train shows that the heavy loads which 
are now common on railroad tracks force the tie into the bal- 
last with the passage of every wheel load. The effect on the 
ballast is a greater or less amount of crushing of the ballast. 
Even tho very hardest grades of broken stone are more or less 
crushed by grinding against each other during the passage of a 
train. The softer and weaker forms of ballast are ground up 
much more quickly. One result is the formation of a fine dust 
which interferes with the proper drainage of water through the 
ballast. A second result is the compression of the ballast imme- 
idiately under the tie into the sub-soil. In a comparatively 
short time a hole is formed under the tie which acts virtually 
like a pump. With every rise and fall of the tie under each 
wheel load, the tie actually pumps the water from the surround- 
ing ballast and sub-soil into these various holes. When the 
ballast is of such a character that the water does not drain 
through it easily, the water will settle in these holes long t>nough 
to seriously deteriorate the ties. When the track beco^nes so 
much out of line or level, or so loose that it needs to be tamped 
up, the process of tamping has practically the effect of deepen- 



§ 198. BALLAST. 231 

ing the amount of ballast immediately under the tie, while the 
sub-soil is forced up between the ties. A longitudinal section 
of the sub-soil of a track which has been frequently tamped 
generally has a saw-tooth appearance, and the sub-soil, instead 
of being a uniform line, has a high spot between each tie, while 
the ballast is considerably below its normal level immediately 
under the tie. 

The variation in the traffic on railroads has caused the Amer- 
ican Railway Engineering and Maintenance of Way Association 
to divide railroads into three classes with respect to the stand- 
ards of construction which should be adopted for ballasting, 
as well as other details of construction. The three classes are 
as follows (quoted from the Association Manual): 

"Class 'A' shall include all districts of a railway having more 
than one main track, or those districts of a railway having a 
single main track with a traffic that equals or exceeds the follow- 
ing: 

Freight-car mileage passing over districts per year per 

mile 150,000 

or, 
Passenger-car mileage per annum per mile of district. . . 10,000 

with maximum speed of passenger-trains of 50 miles per hour. 
"Class 'B' shall include all districts of a railway having a 
single main track with a traffic that equals or exceeds the 
following: 

Freight-car mileage passing over districts per year per 

mile 50,000 

or, 
Passenger-car mileage per annum per mile of district . . 5,000 

with maximum speed of passenger-trains of 40 miles per hour. 

"Class 'C shall include all districts of a railway not meeting 
the traffic requirements of Classes 'A' or 'B/" 

The classification was adopted on the consideration that 
quality of traffic as well as mere tonnage should determine 
the classification of a railroad. For example, it is considered 
that a road which operates a train at a speed of 50 miles an 
hour should adopt the first class of Class "A" standards, even 
though there is but one train per day on that railroad. It 



232 



RAILROAD CONSTRUCTION. 



§ 198. 



likewise means that any road whose traffic makes necessary the 
construction of a regular double track should adopt the first- 
class specifications. 



^i-12-|<— 3'3^ J ^lope y2 to the foot " 

Slope IH to 1- I ^— ^ -' -^-^- -^ 



^ _1_. 




K' ■ 

Froyide drains where needed 



CRUSHED ROCK AND SLAG 



Select coarse stone 
for end of drain 



/ Sod- 




t< 



-10 0- 



i^2^|< 4 '3^^ ^! 

Slope|3tol- 




-.f- 



^r^^^'/y^^^^~y//. 






±=X 



Provide drains where needed Select coarse stone 

for. end of drain 

GRAVEL, CINDERS, CHATS, ETC., 



Sod 




^lope 3 to 1 



GRAVEL, CINDERS, CHATSj EXCj, 
Fig. 106. — Cross-sections op Ballast for Class "A ' Roads. 

In Fig. 106 are shown a series of cross-sections which were 
recommended by that association for Class ^^A'' traffic. It 



§ 198. BALLAST. 233 

should be noticed that in each case the cross-section of the 
roadbed from shoulder to shoulder of the roadbed is 20 feet 
plus the space between track centers for double track if any. 
The width of side ditches is merely added to that of the roadbed. 
The clear thickness of the ballast underneath the ties is made 
12 inches, but even this should be considered as the minimum 
depth and is recommended for use only on the firmest, most 
substantial and well-drained subgrades. The slope of J inch 
to the foot from the center of the track to the end of the tie, 
which is common to all the cross-sections, is designed with the 
idea of allowing a clear space of 1 inch underneath the rail. 
The ballast is then rounded off on a curve of 4 feet radius and 
finally reaches the subsoil on a slope which is H:l for broken 
stone, and 3 : 1 for all other materials. The flat slope adopted 
for gravel, etc., which adds considerably to the required width 
of roadbed, has been so designed in order that the considerable 
mass of material at the ends of the ties shall be better able to 
hold the track in place laterally. The sod on the embank- 
ment over the shoulder of the roadbed up to within 12 inches 
of the edge of the ballast is strongly recommended on account 
of the protection it affords to the shoulder of the roadbed. 
It should be noticed that the latest decision of that associa- 
tion regarding the form of subgrade is that the subgrade should 
be made level and not crowned, as suggested and discussed in 
§63. 

In Fig. 107 are shown a series of cross-sections for various 
classes of ballast for railroads that belong to Class ^'B." It 
may be noted that the thickness of the ballast under the tie 
is 9 inches for this class. The width of roadbed between the 
shoulders, recommended for Class "B*' is 16 feet. As before, 
the width of the ditches is supposed to be added to this width. 
It should be noted that when using cementing gravel and chert 
the slope of 3:1 is made to begin at the bottom of the tie in- 
stead of at a point about 2 inches below the top of the tie. 
This is done in order to prevent water from accumulating 
around the end of the tie in a material which is less permeable 
than the other forms of ballast. 

In Fig. 108 are shown two cross-sections for ballast for roads 
belonging to Class '^C' On roads of this class it is assumed 
that crushed rock will not be used for ballast. The width of 



234 



RAILROAD CONSTRUCTION. 



§198. 



roadbed between shoulders is 14 feet, while the depth of ballast 
underneath the tie is 6 inches. 

It should be noticed that the above sections issued by the 
association do not include any cross-section which is recom- 
mended when no special ballast is used other than the natural 




Crushed rock and slag 




Gravel, cinders, chats, etc 




Cementing gravel and chert. 
Fi». 107. — Cross-sections of Ballast for Class "B" Roads. 

soil. In such a case a cross-section very similar to the sec* 
tions shown for cementing gravel and chert should be used. The 
essential feature of such a section is that the soil, which is 
probably not readily permeable, should be kept away from 
the ends of the ties. Specifications for the placing of mud 
ballast, as well as other forms of ballast, have frequently speci- 
fied that the ballast should be crowned about 1 inch above the 
level of the tops of the ties in the center of the track. This 



199. 



BALLAST. 



235 



feature of any cross-section, although proposed, was rejected 
by the association, in spite of the fact that when a tie is so 
imbedded it certainly will have a somewhat greater holding 
power in the ballast. 



k 



11 / // 



-t'o- 






Slope \i to the foot 




Gravel, cinders, chats, etc 




Cementing gravel and chert. 
Fig. 108. — Cross-sections of Ballast for Class "C" Roads. 



199. Methods of laying ballast. The cheapest method of 
laying ballast on new roads is to lay ties and rails directly on 
the prepared subgrade and run a construction train over the 
track to distribute the ballast. Then the track is lifted up until 
sufficient ballast is worked under the ties and the track is prop- 
erly surfaced. This method, although cheap, is apt to injure 
the rails by causing bends and kinks, due to the passage of 
loaded construction trains when the ties are very unevenly and 
roughly supported, and the method is therefore condemned and 
prohibited in some specifications. The best method is to draw 
in carts (or on a contractor's tempora^-y track) the ballast that 
is required under the level of the bottom of the ties. Spread 
this ballast carefully to the required surface. Then lay the ties 
and rails, which will then have a very fair surface and uniform 
support. A construction train can then be run on the rails and 
distribute sufficient additional ballast to pack around and 
between the ties and make the required cross-section. 

The necessity for constructing some lines at an absolute 



236 RAILROAD CONSTRUCTION. § 200. 

minimum of cost and of opening them for traffic as soon as 
possible has often led to the policy of starting traffic when 
there is little or no ballast — perhaps nothing more than a mere 
tamping of the natural soil under the ties. When this is done 
ballast may subsequently be drawn where required by the train- 
load on flat cars and unloaded at a minimum of cost by means 
of a '^plough/' The plough has the same width as the cars and 
is guided either by a ridge along the center of each car or by 
short posts set up at the sides of the cars. It is drawn from one 
end of the train to the other by means of a cable. The cable is 
sometimes operated by means of a small hoisting-engine car- 
ried on a car at one end of the train. Sometimes the locomo- 
tive is detached temporarily from the train and is run ahead 
with the cable attached to it. 

200. Cost. The cost of ballast in the track is quite a variable 
item for different roads, since it depends (a) on the first cost of 
the material as it comes to the road, (6) on the distance from 
the source of supply to the place where it is used, and (c) on 
the method of handling. The first cost of cinder or slag is 
frequently insignificant. A gravel-pit may cost nothing except 
the price of a little additional land beyond the usual limits of 
the right of way. Broken stone will usually cost $1 or more 
per cubic yard. If suitable stone is obtainable on the com- 
pany's land, the cost of blasting and breaking should be some- 
what less than this. The cost of hauling will depend on the 
distance hauled, and also, to a considerable extent, on the limi- 
tations on the operation of the train due to the necessity of keep- 
ing out of the way of regular trains. There is often a needless 
waste in this way. The *^mud train" is considered a pariah and 
entitled to no rights whatever, regardless of the large daily cost 
of such a train and of the necessary gang of men. The cost of 
broken-stone ballast in the track is estimated at $1.25 per cubic 
yard. The cost of gravel ballast is estimated at 60 c. per cubic 
yard in the track. The cost of placing and tamping gravel 
ballast is estimated at 20* c. to 24 c. per cubic yard, for cinders 
12 c. to 15 c. per cubic yard. The cost of loading gravel on 
cars, using a steam-shovel, is estimated at 6 c. to 10 c. per 
cubic yard.* 

♦ Beport Roadmasters* Association. 1885. 



CHAPTER VIII. 
TIES, 
AND OTHER FORMS OF RAIL SUPPORT. 

201. Various methods of supporting rails. It is necessary 
that the rails shall be sufficiently supported and braced, so that 
the gauge shall be kept constant and that the rails shall not be 
subjected to excessive transverse stress. It is also preferable 
that the rail support shall be neither rigid (as if on solid rock) 
nor too yielding, but shall have a uniform elasticity throughout. 
These requirements are more or less fulfilled by the following 
methods. 

(a) Longitudinals. Supporting the rails throughout their 
entire length. This method is very seldom used in this country 
except occasionally on bridges and in terminals when the 
longitudinals are supported on cross-ties. In § 224 will be 
described a system of rails, used to some extent in Europe, 
having such broad bases that they are self-supporting on the 
ballast and are only connected by tie-rods to maintain the gauge. 

(b) Cast-iron "bowls" or "pots." These are castings resem- 
bling large inverted bowls or pots, having suitable chairs on 
top for holding and supporting the rails, and tied together with 
tie-rods. They will be described more fully later (§ 223). 

(c) Cross- ties of metal or wood. These will be discussed in 
the following sections, 

202. Economics of ties. The true cost of ties depends on the 
relative total cost of maintenance for long periods of time. The 
first cost of the ties delivered to the road is but one item in the 
economics of the question. Cheap ties require frequent renew- 
als, which cost for the labor of each renewal practically the 
same whether the tie is of oak or of hemlock. Cheap ties make 
a poor roadbed which will require more track labor to keep even 
in tolerable condition. The roadbed will require to be disturbed 
so frequently on account of renewals that the ties never get an 
opportunity to get settled and to form a smooth roadbed for any 
length of time. Irregularity in width, thickness, or length of 
ties is especially detrimental in causing the ballast to act and 
wear unevenly. The life of ties has thus a more or less direct 
influence on the life of the rails, on the wear of rolling stock, and 
on the speed of trains. These last items are not so readily 
reducible to dollars and cents, but when it can be shown that 
the total cost, for a long period of time, of several renewals of 

237 



238 



RAILROAD CONSTRUCTION. 



§ 202. 



cheap ties, with all the extra track labor involved, is as great as 
or greater than that of a few renewals of durable ties, then there 
is no question as to the real economy. In the following dis- 
cussions of the merits of untreated ties (either cheap or costly), 
chemically treated ties, or metal ties, the true question is there- 
fore of the ultimate cost of maintaining any particular kind of 
ties for an indefinite period, the cost including the first cost of 
the ties, the labor of placing them and maintaining them to 
surface, and the somewhat uncertain (but not therefore non- 
existent) effect of frequent renewals on repairs of rolling stock, 
on possible speed, etc. 

WOODEN TIES. 

203. Choice of wood. This naturally depends, for any partic- 
ular section of country, on the supply of wood which is most 
readily available. The woods most commonly used, especially 
in this country, are oak and pine, oak being the most durable 
and generally the most expensive. Redwood is used very ex- 
tensively in California and proves to be extremely durable, so 
far as decay is concerned, but it is very soft and is much injured 
by "rail-cutting.'' This defect is being partly remedied by the 
use of tie-plates, as wdll be explained later. Cedar, chestnut, 
hemlock, and tamarack are frequently used in this country. In 
tropical countries very durable ties are frequently obtained from 
the hard woods peculiar to those countries. According to a 
bulletin of the U. S. Department of Agriculture (Forestry 
service. No. 124) the number and value of the cross-ties used 
by the steam and street railroads of the United States during 
the year 1906, was as follows: 



Kind of wood. 


Number of 
ties. 


Per cent. 


Total value. 


Aver, value 


Oaks 


45,857,874 

18,841,210 

7,248.562 

7,083.442 

6,588,975 

5,104,496 

3,969,605 

2,576.859 

2.058,198 

1,248.629 

554.738 

373.387 

1,828,067 


44.1 
18.3 
7.1 
6.9 
6.4 
5.0 
3.9 
2.5 
2.0 
1.2 
0.5 
0.3 
1.8 


$23,278,052 

9,567,745 

3,010,392 

3.310,116 

2,995,942 

1,862,135 

1,698,027 

889,561 

582,968 

536,172 

210,818 

151,052 

726,144 


$0 51 


Southern pines 

Douglas fir 


.51 
42 


Cedar 


.47 


Chestnut 


.49 


Cypress 


.36 


Western pine 

Tamarack 


.43 
.35 


Hemlock 


.28 


Redwood 


.43 


Lodgepole pine 

White pine 


.38 
.40 


All others 


.40 






Total 


102,834,042 


100.0 ' $48,819,124 


.?0 . 47 



§ 205. TIES. 239 

The limitations of timber supply have somewhat dimin- 
ished the use of oak and increased the use of the softer woods 
in recent years. 

204. Durability. The durability of ties depends on the cli- 
mate; the drainage of the ballast; the volume, weight, and 
speed of the traffic; the curvature, if any; the use of tie-plates; 
the time of year of cutting the timber; the age of the timber 
and the degree of its seasoning before placing in the track; the 
nature of the soil in which the timber is grown; and, chiefly, 
on the species of w^ood employed. The variability in these 
items will account for the discrepancies in the reports on the life 
of various woods used for ties. 

White oak is credited with a life of 5 to 12 years, depending 
principally on the traffic. It is both hard and durable, the 
hardness enabling it to withstand the cutting tendency of the 
rail-flanges, and the durability enabling it to resist decay. Pine 
and redwood resist decay very well, but are so soft that they are 
badly cut by the rail-flanges and do not hold the spikes very 
well, necessitating frequent respiking. Since the spikes must 
be driven within certain very limited areas on the face of each 
tie, it does not require many spike-holes to "spike-kilP' the 
tie. On sharp curves, especially with heavy traffic, the wheel- 
flange pressure produces a side pressure on the rail tending to 
overturn it, which tendency is resisted by the spike, aided some- 
times by rail-braces. Whenever the pressure becomes too great 
the spike will yield somewhat and will be slightly ^vithdrawn. 
The resistance is then somew^hat less and the spike is soon so 
loose that it must be redriven in a new hole. If this occurs 
very often, the tie may need to be replaced long before any decay 
has set in. When the traffic is very light, the wood very dura- 
ble, and the climate favorable, ties have been known to last 
25 years. 

205. Dimensions. The usual dimensions for the best roads 
(standard gauge) are 8' to 9' long, 6" to 7" thick, and 8" to 
10" wide on top and bottom (if they are hewed) or 8" to 9" 
wide if they are sawed. For cheap roads and light traffic the 
length is shortened sometimes to 7' and the cross-section also 
reduced. On the other hand a very few roads use ties 9' 6" long. 

Two objections are urged against sawed ties: first, that the 
grain is torn by the saw, leaving a woolly surface which induces 
decay; and secondly, that, since timber is not perfectly straight- 



240 RAILROAD CONSTRUCTION. § 205. 

grained, some of the fibers are cut obliquely, exposing their ends, 
which are thus liable to decay. The use of a ^'planer-saw'' ob- 
viates the first difficulty. Chemical treatment of ties obviates 
both of these difficulties. Sawed ties are more convenient to 
handle, are a necessity on bridges and trestles, and it is even 
claimed, although against commonly received opinion, that 
actual trial has demonstrated that they are more durable than 
hewed ties. 

206. Spacing. The spacing is usually 14 to 16 ties to a 30- 
foot rail. This number is sometimes reduced to 12 and even 
10, and on the other hand occasionally increased to 18 or 20 by 
employing narrower ties. There is no economy in reducing the 
number of ties ver}^ much, since for any required stiffness of 
track it is more economical to increase the number of supports 
than to increase the weight of the rail. The decreasing cost of 
rails and the increasing cost of ties have materially changed the 
relation between number of ties and weight of rail to produce a 
given stiffness at minimum cost, but many roads have found it 
economical to employ a large number of ties rather than increase 
the weight of the rail. On the other hand there is a practical 
limit to the number that may be employed, on account of the 
necessary space between the ties that is required for proper 
tamping. This width is ordinarily about twice the width of the 
tie. At this rate, with light ties 6'' wide and with 12'' clear 
space, there would be 20 ties per 30-foot rail, or 3520 per mile. 
The smaller ties can generally be bought much cheaper (propor- 
tionately) than the larger sizes, and hence the economy. 

Track instructions to foremen generally require that the 
spacing of ties shall not be uniform along the length of any 
rail. Since the joint is generally the weakest part of the rail 
structure, the joint requires more support than the center of the 
rail. Therefore the ties are placed with but 8^' or 10" clear 
space between them at the joints, this applying to 3 or 4 ties at 
each joint; the remaining ties, required for each rail length, are 
equally spaced along the remaining distance. 

207. Specifications. The specifications for ties are apt to 
include the items of size, kind of wood, and method of construc- 
tion, besides other minor directions about time of cutting, sea- 
soning, delivery, quality of timber, etc. 

(a) Size. The particular size or sizes required wall be some- 
what as indicated in § 205. 



§ 208. TIES. 241 

(b) Kind of wood. When the kind or kinds of wood are 
specified^ the most suitable kinds that are available in that 
section of country are usually required. 

(c) Method of construction. It is generally specified that the 
ties shall be hewed on two sides; that the two faces thus made 
shall be parallel planes and that the bark shall be removed. It 
is sometimes required that the ends shall be sawed off square; 
that the timber shall be cut in the winter (when the sap is down) ; 
and that the ties shall be seasoned for six months These last 
specifications are not required or lived up to as much as their 
importance deserves. It is sometimes required that the ties shall 
be delivered on the right of way, neatly piled in rows, the alter- 
Date rows at right angles, piled if possible on ground not lower 
than the rails and at least seven feet away from them, the lower 
row of ties resting on tw^o ties which are themselves supported 
so as to be clear of the ground. 

(d) Quality of timber. The usual specifications for sound 
timber are required, except that they are not so rigid as for a 
better class of timber work The ties must be sound, reason- 
ably straight-grained, and not very crooked — one test being that 
a line joining the center of one end with the center of the middle 
shall not pass outside of the other end. Splits or shakes, espe- 
cially if severe, should cause rejection. 

Specifications sometimes require that the ties shall be cut 

from single trees, making 
what is known as ^'pole 
ties" and definitely con- 
demning those w^hich are 

„ ^^^ „ „ cut or split from larger 

Fia, 109. — Methods op cutting Ties. , , f . , .. f i 

trunks, givmg two slab 

ties" or four " quarter ties" for each cross-section, as is illustrated 
in Fig. 109. Even if pole ties are better, their exclusive use 
means the rapid destruction of forests of young trees. 

2o8. Regulations for laying and renewing ties. The regula- 
tions issued by railroad companies to their track foremen will 
generally include the following, in addition to directions regard- 
ing dimensions, spacing, and specifications given in §§ 204-207. 
When hew^n ties of somew^hat variable size are used, as is fre- 
quently the case, the largest and best are to be selected for use 
as joint ties. If the upper surface of a tie is found to be warped 
(contrary to the usual specifications) so that one or both rails do 




POLE TIE. SLAB TIE. QUARTER TIE. 



242 RAILROAD CONSTRUCTION. § 208. 

not get a full bearing across the whole width of the tie, it musfc 
be adzed to a true surface along its whole length and not merely 
notched for a rail-seat. When respiking is necessary and spikes 
have been pulled out, the holes should be immediately plugged 
with '^ wooden spikes," which are supplied to the foreman for 
that express purpose, so as to fill up the holes and prevent the 
decay which would otherwise take place when the hole becomes 
filled with rain-water. Ties should always be laid at right angles 
to the rails and never obliquely Minute regulations to prevent 
premature rejection and renewal of ties are frequently made. It 
is generally required that the requisitions for renewals shall be 
made by the actual count of the individual ties to be renewed 
instead of by any wholesale estimates. It is unwise to have ties 
of widely variable size, hardness, or durability adjacent to each 
other in the track, for the uniform elasticity, so necessary for 
smooth riding, will be unobtainable under those circumstances. 

209. Cost of ties. When railroads can obtain ties cut by 
farmers from woodlands in the immediate neighborhood, the 
price will frequently be as low as 20 c for the smaller sizes, 
running up to 50 c for the larger sizes and better qualities, espe- 
cially when the timber is not very plentiful Sometimes if a 
railroad cannot procure suitable ties from its immediate neigh- 
borhood, it will find that adjacent railroads control all adjacent 
sources of supply for their own use and that ties can only be 
procured from a considerable distance, with a considerable added 
cost for transportation. First-class oak ties cost about 75 to 80 c. 
and frequently much more Hemlock ties can generally be 
obtained for 35 c. or less. 

PRESERVATIVE PROCESSES FOR WOODEN TIES. 

210. General principle. Wood has a fibrous cellular struc- 
ture, the cells being filled with sap or air. 'J he woody fiber is 
but little subject to decay unless the sap undergoes fermentation. 
Preservative processes generally aim at removing as much of the 
water and sap as possible and filling up the pores of the wood 
with an antiseptic compound The most common methods (ex- 
cept one) all agree in this general process and only differ in the 
method employed to get rid of the sap and in the antiseptic 
chemical with which the fibers are filled One valuable feature 
of these processes lies in the fact that the softer cheaper woods 



§ 212. TIES. 243 

(such as hemlock and pine) are more readily treated than are the 
harder woods and yet will produce practically as good a tie as a 
treated hard-wood tie and a very much better tie than an un- 
treated hard-wood tie. The various processes will be briefly 
described, taking up first the process which is fundamentally 
different from the others, viz., vulcanizing. 

211. Vulcanizing. The process consists in heating the timber 
to a temperature of 300° to 500° F. in a cylinder, the air being 
under a pressure of 100 to 175 lbs. per square inch. By this 
process the albumen in the sap is coagulated, the water evapo- 
rated, and the pores are partially closed by the coagulation of 
the albumen. It is claimed that the heat sterilizes the wood and 
produces chemical changes in the wood which give it an antisep- 
tic character. It was once very extensively used on the ele- 
vated lines of New York City, but the process has now been 
abandoned as unsatisfactor3^ 

212. Creosoting. — This process consists in impregnating the 
wood with wood-creosote or with dead oil of coal-tar. Wood- 
creosote is one of the products of the destructive distillation of 
wood — usually long-leaf pine. Dead oil of coal-tar is a prod- 
uct of the distillation of coal-tar at a temperature between 480° 
and 760° F. It would require about 35 to 50 pounds of creo- 
sote to completely fill the pores of a cubic foot of wood But 
it would be impossible to force such an amount into the wood, 
nor is it necessary or desirable. About 10 pounds per cubic 
foot, or about 35 pounds per tie, is all that is necessary. For 
piling placed in salt water about 18 to 20 pounds per cubic foot 
is used, and the timber is then perfectly protected against the 
ravages of the teredo navalis. To do the work, long cylinders, 
which may be opened at the ends, are necessary. Usually the 
timbers are run in and out on iron carriages running on rails 
fastened to braces on the inside of the cylinder. When the load 
has been run in, the ends of the cylinder are fastened on. The 
water and air in the pores of the wood are first drawn out by 
subjecting the wood alternately to steam- pressure and to the 
action of a vacuum-pump. This is continued for several hours. 
Then, after one of the vacuum periods, the cylinder is filled 
with creosote oil at a temperature of about 170° F. The pumps 
are kept at work until the pressure is about 80 to 100 pounds 
per square inch, and is maintained at this pressure from one to 
two hours according to the size of the timber. The oil is then 



244 RAILROAD CONSTRUCTION. § 213. 

withdrawTi, the cylinders opened, the train pulled out and an- 
other load made up in 40 to 60 minutes. The average time re- 
quired for treating a load is about 18 or 20 hours, the absorption 
about 10 or 11 pounds of oil per cubic foot, and the cost (1894) 
from $12.50 to $14.50 per thousand feet B. AL 

213. Burnettizing (chloride-of-zinc process). This process is 
very similar to the creosoting process except that the chemical is 
chloride of zinc, and that the chemical is not heated before use. 
The preliminary treatment of the wood to alternate vacuum and 
pressure is not continued for quite so long a period as in the 
creosoting process. Care must be taken, in using this process, 
that the ties are of as uniform quality as possible, for seasoned 
ties Avill absorb much more zinc chloride than unseasoned (in the 
same time), and the product will lack uniformity unless the sea- 
soning is uniform. The A., T. & S. Fe R. R. has works of its 
own at which ties are treated by this process at a cost of about 
25 c. per tie. The Southern Pacific R R. also has works for 
burnettizing ties at a cost of 9.5 to 12 c per tie The zinc- 
chloride solution used in these works contains only 1.7% of zinc 
chloride instead of over 3% as used in the Santa Fe works, which 
perhaps accounts partially for the great difference in cost per tie. 
One great objection to burnettized ties is the fact that the chem- 
ical is somewhat easily washed out, when the wood again be- 
comes subject to decay Another objection, which is more 
forcible with respect to timber subject to great stresses^ as in 
trestles, than to ties, is the fact that when the solution of zinc 
chloride is made strong (over 3%) the timber is made very brittle 
and its strength is reduced. The reduction in strength has been 
shown by tests to amount to ^ to ^^ of the ultimate strength, 
and that the elastic limit has been reduced by about ^. 

214. Kyanizing (bichloride-of- mercury or corrosive-sublimate 
process). This is a process of "steeping" It requires a much 
longer time than the previously described processes, but does not 
require such an expensive plant. Wooden tanks of sufficient 
size for the timber are all that is necessary. The corrosive subli- 
mate is first made into a concentrated solution of one part of 
chemical to six parts of hot water When used in the tanks this 
solution is weakened to 1 part in 100 or 150 The wood will 
absorb about 5 to G.5 pounds of the bichloride per 100 cubic 
feet, or about one pound for each 4 to 6 ties. The timber is 
allowed to soak in the tanks for several days, the general rule 



§ 215. TIES. 245 

being about one day for each inch of least thickness and one day 
over — which means seven days for six-inch ties, or thirteen (to 
fifteen) days for 12" timber (least dimension). The process is 
somewhat objectionable on account of the chemical being such a 
virulent poison, workmen sometimes being sickened by the fumes 
arising from the tanks. On the Baden railway (Germany) 
kyanized ties last 20 to 30 years. On this railway the wood is 
always air-dried for two weeks after impregnation and before 
being used, which is thought to have an important effect on its 
durability. The solubility of the chemical and the liability of 
the chemical washing out and leaving the wood unprotected is 
an element of weakness in the method. 

215. Wellhouse (or zinc- tannin) process. The last two 
methods described (as well as some others employing similar 
chemicals) are open to the objection that since the wood is im- 
pregnated with an aqueous solution, it is liable to be washed out 
very rapidly if the wood is placed under water, and will even 
disappear, although more slowly, under the action of moisture 
and rain. Several processes have been proposed or patented to 
prevent this. Many of them belong to one class, of which the 
Wellhouse process is a sample. By these processes the timber 
is successively subjected to the action of two chemicals, each 
individually soluble in water, and hence readily impregnating 
the timber, but the chemicals when brought in contact form in- 
soluble compounds which cannot be washed out of the wood- 
cells. By the Wellhouse process, the wood is first impregnated 
with a solution of chloride of zinc and glue, and is then subjected 
to a bath of tannin under pressure. The glue and tannin com- 
bine to form an insoluble leathery compound in the cells, which 
will prevent the zinc chloride from being washed out. It is 
being used by the A., T. &. S. Fe R. R., their works being 
located at Las Vegas, New Mexico, and also by the Union 
Pacific R. R. at their works at Laramie, W^o. In 1897 Mr. J. 
M. Meade, a resident engineer on the A., T. & S. Fe, exhibited 
to the Roadmasters Association of America a piece of a tie treated 
by this process which had been taken from the tracks after 
nearly 13 years^ service. The tie was selected at random, was 
taken out for the sole purpose of having a specimen, and was 
still in sound condition and capable of serving many 3^ears longer. 
The cost of the treatment was then quoted as 13 c. per tie. 



246 RAILROAD CONSTRUCTION. § 216. 

It was claimed that the treatment trebled the life of the tie 
besides adding to its spike-holding power. 

In spite of this apparently favorable showing, the process 
was abandoned on the A. T. & S. F. R. R. in 1898 on the 
ground that the results did not justify the added expense. 

2i6. Cost of Treating. The cost of treating ties by the vari- 
ous methods has been estimated as follows * — assuming that 
the plant was of sufficient capacity to do the work economi- 
cally: creosoting, 25 c. per tie; vulcanizing, 25 c. per tie; 
burnettizing (chloride of zinc), 8.25 c. per tie; kyanizing (steep- 
ing in corrosive sublimate), 14.6 c. per tie; Wellhouse process 
(chloride of zinc and tannin), 11.25 c. per tie. These estimates 
are only for the net cost at the works and do not include the 
cost of hauling the ties to and from the works, which may mean 
5 to 10 c. per tie. Some of these processes have been installed 
on cars which are transported over the road and operated where 
most convenient. An estimate made in 1907 by Prof. Gellert 
Alleman on the cost of treating ties, each assumed to have a 
volume of 3 cubic feet, the cost ^'not including royalty on pa- 
tents, profit, interest, or depreciation, all of which vary widely 
at the various plants," is as follows: 

Zinc chloride 16 cents 

" " and creosote 27 " 

Creosote, 10 pounds to the cubic foot . . . 55 ** 

The very grert increase in these prices, especially for creosot- 
ing, is due to the enormous increase in late years in the con- 
sumption and in the price of creosote. 

217. Economics of treated ties. The fact that treated ties are 
not universally adopted is due to the argument that the added 
life of the tie is not worth the extra cost. If ties can be bought 
for 25 c, and cost 25 c. for treatment, and the treatment only 
doubles their life, there is apparently but little gained except 
the work of placing the extra tie in the track, which is more 
or less offset by the interest on 25 c. for the life of the untreated 
tie, and the larger initial outlay makes a stronger impression on 
the mind than the computed ultimate economy. But when 
(utilizing some statistics from the Pittsburg, Ft. Wayne & 

* Bull. No. 9. U. S. Dept. of Agric, Div. of Forestry. App. No. 1, by 
Henry Flad. 



§ 217. TIES. 247 

Chicago Railroad) it is found that white oak ties laid in rock 
ballast had a life of 10.17 years, and that hemlock ties treated 
with the zinc-tannin process and laid in the same kind of ballast 
lasted 10.71 years, then the economy is far more apparent. 
Unfortunately no figures were given for the cost of these ties 
nor for the cost of the treatment; but if we assume that the 
white oak ties cost 75 c. and the hemlock ties 35 c. plus 20 c. 
for treatment, there is not only a saving of 20 c. on each tie, 
but also the advantage of the slightly longer life of the treated 
tie. In the above case the total life of the two kinds of ties 
is so nearly the same that we may make an approximation of 
their relative worth by merely comparing the initial cost; but 
usually it is necessary to compare the value of two ties one 
of which may cost more than the other, but will last considerably 
longer. The mathematical comparison of the real value of 
two ties under such conditions may be developed as follows: 
The real cost of a tie, or any other similar item of constructive 
jwork, is measured by the cost of perpetually maintaining that 
item in proper condition in the structure. It will be here 
assumed that the annual cost of the trackwork, which is assign- 
able to the tie, is the same for all kinds of ties, although the 
difference probably lies in favor of the more expensive and 
most durable ties. By assuming this expense as constant, the 
remaining expense may be considered as that due to the cost 
of the new ties whenever necessary, plus the cost of placing 
them in the track. We also may combine these two items 
in one, and consider that the cost of placing a tie in the track, 
which we will assume at the constant value of 20 c. per tie, 
regardless of the kind of tie, is merely an item of 20 c. in the 
total cost of the tie. We will assume that T^ is the present 
cost of a tie, the cost including the preservative treatment if 
any, and the cost of placing in the track. The tie is assumed 
to last n years. At the end of n years another tie is placed 
in the track, and, for lack of more precise knowledge, we will 
assume that this cost Tj equals T^, The "present worth" 
of 7^2 is the sum which, placed at compound interest, would 
equal Tj at the end of n years, and is expressed by the quantity 

T 
^ ^, in which r equals the rate of interest. Similarly at 



(1+r) 

the end of 2n years we must expend a sum T^ to put in the third 

tie, and the present worth of the cost of that third tie is ex- 



248 RAILROAD CONSTRUCTION. § 217. 

T 
pressed by the fraction . %9 ~' We may similarly express 

the present worths of the cost of ties for that particular spot 
for an indefinite period. The sum of all these present worths 
is given by the sum of a converging series and equals (assuming 

that all the T^s are equal) . . — . But instead of laying 

aside a sum of money which will maintain a tie in that par- 
ticular place in perpetuity, we may compute the annual sum 
which must be paid at the end of each year, which would be 
the equivalent. We will call that annual payment A, and 
then the present worths of all these items are as follows: 

For the first payment YVa^S^ 

For the second payment — r^: 

For the third payment ■ ^ 3 ; 

For the nth payment ^. 

^ •^ (l-fr)n 

After the next tie is put in place we have the present worths 
of the annual payments on the second tie, of which the first 
one would be 

For the (n+1) payment {i+r)(n+i y 

Similarly after x ties have been put in place the last pay- 
ment for the x tie would have a present worth -; . The 

^ (l+r)na;- 

sum of all these present worths is represented by the sum of 
a converging series and equals the very simple expression — • 

But since the sum of the present worths of these annual pay- 
ments must equal the sum of the present worths of the payments 
made at intervals of n years, we may place these two summa- 
tions equal to each other, and say that 

(l+r)^-l 



§ 217. TIES. 249 

Values of A for various costs of a tie T on the basis that r 
equals 5% have been computed and placed in Table XXXIV. 
To illustrate the use of this table, assume that we are comparing 
the relative values of two ties, both untreated, one of them 
a white oak tie which will cost, say 75 c, and will last twelve 
years, the other a yellow pine tie which will cost, say 35 c, 
and will last six years. Assuming a charge for each case of 
20 c. for placing the tie in the track, we have as the annual 
charge against the white oak tie, which costs 95 c. in the track, 
10.72 c. The pine tie, costing 55 c. in the track and lasting 
six years, will be charged with an annual cost of 10.48 c, which 
shows that the costs are practically equal. It is probably 
true that the track work for maintaining the white oak would 
be less than that for the pine tie, but since the initial cost of 
the pine tie is less than that of the oak tie, it would probably 
be preferred in this case, especially if money was difficult to 
obtain. It may be interesting to note that if a comparison is 
made from a similar table which is computed on the basis of 
compounding the money at 4% instead of 5%, the annual 
charges would be 10.13 and 10.49 c. for the oak and pine ties 
respectively, thus showing that when money is "easier" the 
higher priced tie has the greater advantage. 

Example 2. Considering again the comparison previously 
made of a white oak untreated tie which was assumed to cost 
75 c, and a hemlock treated tie, which cost 35 c. for the tie 
and 20 c. for the treatment, the total costs of these ties laid 
in the track would therefore be 95 c. and 75 c. respectively. 
These ties had practically the same life (10.17 and 10.71 years), 
but in order to use the table, we will call it ten years for each 
tie. The annual charge against the oak tie would therefore 
be 12.30 c, while that against the hemlock tie would be 9.72 c. 
This gives an advantage in the use of the treated tie of 2.58 c. 
per year, which capitalized at 5% would have a capitalized 
value of 51.6 c. 

The Atchison, Topeka and Santa Fe R. R. has compiled a 
record of treated pine ties removed in 1897, '98, '99, and 1900, 
showing that the average life of the ties removed had been about 
11 years. On the Chicago, Rock Island and Pacific R. R., the 
average life of a very large number of treated hemlock and 
tamarack ties was found to be 10.57 years. Of one lot of 21,850 
ties, 12% still remained in the track after 15 years' exposure. 

It has been demonstrated that much depends on the minor 



250 RAILROAD CONSTRUCTION. § 218. 

details of the process — whatever it may be. As an illustra- 
tion, an examination of a batch of ties, treated by the zinc- 
creosote process, showed 84% in service after 13 years' ex- 
posure; another batch, treated by another contractor by the 
same process (nominally), showed 50% worthless after a service 
of six years. 

METAL TIES. 

2 1 8. Extent of use. In 1894 * there were nearly 35000 miles 
of ''metal track" in various parts of the world. Of this total, 
there were 3645 miles of ''longitudinals" (see § 224), found ex- 
clusively in Europe, nearly all of it being in Germany. There 
were over 12000 miles of ''bowls and plates" (see^ 223), found 
almost entirely in British India and in the Argentine Republic. 
The remainder, over 18000 miles, was laid with metal cross-ties 
of various designs. There were over 8000 miles of metal cross- 
ties in Germany alone, about 1500 miles in the rest of Europe, 
over 6000 miles in British India, nearly 1000 miles in the rest 
of Asia, and about 1500 miles more in various other parts of the 
world. Several railroads in this country have tried various de- 
signs of these ties, but their use has never passed the experi- 
mental stage. These 35000 miles represent about 9% of the 
total railroad mileage of the world — nearly 400000 miles. They 
represent about 17.6% of the total railroad mileage, exclusive of 
the United States and Canada, where they are not used at all, 
except experimentally. In the four years from 1890 to 1894 the 
use of metal track increased from less than 25000 miles to nearly 
35000 miles. This increase was practically equal to the total in- 
crease in railroad mileage during that time, exclusive of the 
increase in the United States and Canada. This indicates a 
large growth in the percentage of metal track to total mileage, 
and therefore an increased appreciation of the advantages to be 
derived from their use. 

219. Durability. The durabilit}^ of metal ties is still far 
from being a settled question, due largely to the fact that the 
best form for such ties is not yet determined, and that a large 
part- of the apparent failures in metal ties have been evidently 
due to defective design. Those in favor of them estimate the 
life as from 30 to 50 years. The opponents place it at not more 

* Bulletin No. 9, U. S. Dept. of Agriculture, Div. of Forestry. 



§ 220. _ TIES. 251 

than 20 years, or perhaps as long as the best of wooden ties. 
UnHke the wooden tie, however, which deteriorates as much 
with time as with usage, the Ufe of a metal tie is more largely a 
function of the traffic. The life of a well-designed metal tie has 
been estimated at 150000 to 200000 trains; for 20 trains per 
day, or say 6000 per year, this would mean from 25 to 33 3'ears. 
20 trains per day on a single track is a much larger number than 
will be found on the majority of railroads. Metal ties are found 
to be subject to rust, especially when in damp localities, such as 
tunnels; but on the other hand it is in such confined localities, 
where renewals are troublesome, that it is especially desirable to 
employ the best and longest -lived ties. Paint, tar, etc., have 
been tried as a protection against rust, but the efficacy of such 
protection is as yet uncertain, the conditions preventing any re- 
newal of the protection — such as may be done by repainting a 
bridge, for example. Failures in metal cross-ties have been 
largely due to cracks which begin at a comer of one of the square 
holes which are generally punched through the tie, the holes 
being made for the bolts by which the rails are fastened to the 
tie. The holes are generally punched because it is cheaper. 
Reaming the holes after punching is thought to be a safeguard 
against this frequent cause of failure. Another method is to 
round the corners of the square punch with a radius of about 
J''. If a crack has already started, the spread of the crack may 
be prevented by drilling a small hole at the end of it. 

220. Form and dimensions of metal cross- ties. Since stabihty 
in the ballast is an essential quality for a tie, this must be accom- 
plished either by turning down the end of the tie or by having 
some form of lug extending downward from one or more points 
of the tie. The ties are sometimes depressed in the center (see 
Plate VI, N. Y. C. & H. R. R. R. tie) to allow for a thick cover- 
ing of ballast on top in order to increase its stability in the 
ballast. This form requires that the ties should be sufficiently 
well tamped to prevent a tendency to bend out straight, thus 
widening the gauge. Many designs of ties are objectionable 
because they cannot be placed in the track ^\'ithout disturbing 
adjacent ties. The failure of many metal cross-ties, otherwise 
of good design, may be ascribed to too light weight. Those 
weighing much less than 100 pounds have proved too light. 
From 100 to 130 pounds weight is being used satisfactorily on 
German railroads. The general outside dimensions are about 



252 RAILROAD CONSTRUCTION. § 220. 

the same as for wooden ties, except as to thickness. The metal 
is generally from V' to I" thick. They are, of course, only made 
of wrought iron or steel, cast iron being used only for "bowls'' or 
'^ plates'' (see § 223). The details of construction for some 
of the most commonly used ties may be seen by a study of 
Plate VI. 

The Carnegie tie is perhaps the only tie whose use on steam 
railroads in this country has passed the experimental stage. 
The Bessemer and Lake Erie R. R. has nearly 100 miles of 
track laid with these ties, and other roads are making extensive 
experiments. One practical difficulty, which is not of course 
insuperable, arises from the common practice of using the rails 
as parts of an electrical circuit for a block-signal system, which 
requires that the rails shall be insulated from each other. This 
requires that these metal ties shall be insulated from the rails. 
A method of insulation which is altogether satisfactory and* 
inexpensive is yet to be determined. It is claimed that, on 
account of the better connection between the rail and the tie, 
there is less wear and more uniform wear to the rail. It is 
also claimed that there is greater lateral rigidity in the rails 
and ties (considered as a structure) and that this decreases the 
track work necessary to maintain alinement. These ties weigh 
19.7 pounds per linear foot, or about 167 pounds for an 8 foot 
6 inch tie. Even at the lowest possible price per pound the 
cost of the tie and its fastenings must be two or three times 
that of the best oak tie with spikes and even tie plates. It 
has been impossible to estimate the probable life of these ties. 
Until a reasonably close estimate of the life of steel ties can 
be determined, no proper comparison can be made of their 
economy relative to that of wooden ties. A study of Table 
XXXIV will show that a tie which costs, say three times as 
much as a cheap tie, must last more than three times as long 
in order that the annual charge against the tie shall be as low 
as that of the cheaper tie. For example, let us assume that 
the cost of a metal tie, laid in the track, is S2. 55 and that it 
will last 20 years. From Table XXXIV we may find that the 
annual charge against S2.55 at 5% for 20 years = (2X8.02) + 
4.41 = 20.45 c. Compared with a tie costing 65 c, plus 20 c. 
for track laying, we find that the cheaper tie will only cost 
19.63 c. per year even if it only lasts 5 years. Of course the 
claimed advantage of better track and less cost for track main- 
tenance, using steel ties, will tend to offset, so far as it is true, 




CARNEGIE STEEL TIE (1905) 



Plate VI. — Some forms of Metal Ties. 
{Between pp. 251 and 252.) 



i^. 



§ 223. TIES. 253 

the disadvantage of the extra cost of the metal tie. Even 
if the extra work per tie amounts to only one-half hour for 
one man in a year, the cost of it, say 6 c., will utterly change 
the relative economics of the two ties. 

221. Fastenings. The devices for fastening the rails to the 
ties should be such that the gauge may be widened if desired on 
curves, also that the gauge can be made true regardless of slight 
inaccuracies in the manufacture of the ties, and also that shims 
may be placed under the rail if necessary during cold weather 
when the tie is frozen into the ballast and cannot be easily 
disturbed. Some methods of fastening require that the base of 
the rail be placed against a lug which is riveted to the tie or 
which forms a part of it. This has the advantage of reducing 
the number of pieces, but is apt to have one or more of the 
disadvantages named above. Metal keys or wooden wedges are 
sometimes used, but the majority of designs employ some form 
of bolted clamp. The form adopted for the experimental ties 
used by the N. Y. C. & H. R. R. R. (see Plate VI) is especiaUy 
ingenious in the method used to vary the gauge or allow for 
inaccuracies of manufacture. Plate VI shows some of the 
methods of fastening adopted on the prinoipal types of ties. 

222. Cost. The cost of metal cross- ties in Germany averages 
about 1.6 c. per pound or about $1.60 for a 100-lb. tie. The ties 
manufactured for the N. Y. C. & H. R. R. R. in 1892 weighed 
about 100 lbs. and cost $2.50 per tie, but if they had been made 
in larger quantities and with the present price of steel the cost 
would possibly have been much lower. The item of freight 
from the place of manufacture to the place where used is no 
inconsiderable item of cost with some roads. Metal cross-ties 
have been used by some street railroads in this country. Those 
used on the Terre Haute Street Railway weigh 60 pounds and 
cost about 66 c. for the tie, or 74 c. per tie with the fastenings. 



223. Bowls or plates. As mentioned before, over 12000 miles 
of railway, chiefly in British India and in the Argentine Repub- 
lic, are laid with this form of track. It consists essentially of 
large cast-iron inverted ^' bowls" laid at intervals under each 
rail and opposite each other, the opposite bowls being tied 
together with tie-rods. A suitable chair is riveted or bolted on 
to the top of each bowl so as to properly hold the rail. Being 



254 RAILROAD CONSTRUCTION. § 224. 

made of cast iron, they are not so subject to corrosion as steel 
or wrought iron. They have the advantage that when old and 
worn out their scrap value is from 60% to 80% of their initial 
cost, while the scrap value of a steel or wrought-iron tie is prac- 
tically nothing. Failure generally occurs from breakage, the 
failures from this cause in India being about 0.4% per annum. 
They weigh about 250 lbs. apiece and are therefore quite expen- 
sive in first cost and transportation charges. There are miles 
of them in India which have already lasted 25 years and are 
still in a serviceable condition. Some illustrations of this form 
of tie are shoAvn in Plate VI. 

224. Longitudinals.* This form, the use of which is con- 
fined almost exclusively to Germany, is being gradually replaced 
on many lines by metal cross-ties. The system generally con- 
sists of a compound rail of several parts, the upper bearing rail 
being very light and supported throughout its length by other 
rails, which are suitably tied together with tie-rods so as to 
maintain the proper gauge, and which have a sufficiently broad 
base to be properly supported in the ballast. One great objec- 
tion to this method of construction is the 
difficulty of obtaining proper drainage espe- 
cially on grades, the drainage having a ten- 
dency to follow along the lines of the rails. 
,^,,, ,. The construction is much more complicated 
on sharp curves and at frogs and switches. 
Another fundamentally different form of 
longitudinal is the Haarman compound '^self-bearing " rail, 
having a base 12" wide and a height of 8", the alternate sections 
breaking joints so as to form a practically continuous rail. 

Some of the other forms of longitudinals are illustrated in 
Plate VI. 

For a very complete discussion of the subject of metal ties, 
see the '^Report on the Substitution of Metal for Wood in 
Railroad Tics" by E. E. Russell Tratman, it being Bulletin 
No. 4, Forestry Di^dsion of the U. S. Dept. of Agriculture. 

* Although the discussion of longitudinals might be considered to be 
long more properly to the subject of Rails, yet the essential idea of all de- 
signs must necessarily be the support of a rail-head on which the rolling \ 
stock may run, and therefore this form, unused in this country, will be 
briefly described here. 



§ 224a. METAL TIES. 255 

224a. Reinforced Concrete Ties. — The wide application of 
reinforced concrete to various structural purposes, combined 
with its freedom from decay, has led to its attempted adoption 
for ties. At present (1908) their use is wholly in the experi- 
mental stage. In the annual Proceedings of the American 
Railway Engineering and Maintenance of Way Association for 
1907 is a report on over a dozen different designs, the most of 
which were shown to be incapable of enduring traffic except on 
sidings. The ties are particularly subject to fracture if struck 
by a derailed car. 

One of the most successful of these ties is the "Buhrer," 
which consists of one-fourth part of a thirty-foot scrap rail, 
which is inverted so that the base forms the seat of the running 
rails. This rail is imbedded in a mass of concrete whose form 
is somewhat like that of a huge ^'pole'' tie. Several thousands 
of these are in use on various roads, but many of them have 
already required renewal and none of them have yet had time 
to show a service which would make them more economical 
than wooden ties. 



CHAPTER TX. 

RAILS. 

225. Early forms. The first rails ever laid were wooden 
stringers which were used on very short tram-roads around coal- 
mines. As the necessity for a more durable rail increased, 
owing chiefly to the invention of the locomotive as a motive 
power, there were invented successively the cast-iron " fish- 
belly '^ rail and various forms of wrought-iron strap rails which 
finally developed into the T rail used in this country and the 
double-headed rail, supported by chairs, used so extensively in 
England. The cast-iron rails were cast in lengths of about 3 
feet and were supported in iron chairs which were sometimes 
set upon stone piers. A great deal of the first railroad track 
of this country was laid with longitudinal stringers of wood 
placed upon cross-ties, the inner edge of the stringers being 
protected by wrought- iron straps. The "bridge" rails were 
first rolled in this country in 1844. The "pear" section was 
an approach to the present form, but was very defective on 
account of the difficulty of designing a good form, of joint. The 
"Stevens" section was designed in 1830 by Col. Robert L. 
Stevens, Chief Engineer of the Camden and Amboy Railroad; 
although quite defective in its proportions, according to the 
present knowledge of the requirements, it is essentially the pres- 
ent form. In 1836, Charles Vignoles invented essentially the 
same form in England; this form is therefore known throughout 
England and Europe as the Vignoles rail. 

226. Present standard forms. The larger part of modern 
railroad track is laid with rails which are either "T" rails or 
the double-headed or " bull-headed " rails w^hich are carried in 
chairs. The double-headed rail vv-as designed with ci symmetri- 
cal form with the idea that after one head had been worn out 
by traffic the rail could be reversed, and that its life would be 
practically doubled. Experience has shown that the wear of the 

256 



§ 226. 



RAILS. 



257 



rail in the chairs is very great; so much so that when one head 
has been worn out by traffic the w^hole rail is generally useless. 




BALT. & OHIO R. R. 
QUINCYR.R. 1843. "BULL-HEAD." 





CAMDEN & AMBOY. 8TEPKEN50.N. "PEAR.'' 

1832. 1838. 




'FISH-BELLY"— CAST IRON. 




reyn.qlds— 1763j. 
Fig. 111. — Early Forms of Rails. 

If the rail is turned over, the worn places, caused by the chairs, 
make a rough track and the rail appears to be more brittle and 
subject to fracture, possibly due to the crystallization that may 
have occurred during the previous usage and to the reversal of 
stresses in the fibers. Whatever the explanation, experience has 
demonstrated the fact. The ''bull-headed" 
rail has the lower head only large enough to 
properly hold the wooden keys with which 
the rail is secured to the chairs (see Fig. 112) 
and furnish the necessary strength. The use 
of these rails requires the use of two cast- 
iron chairs for each tie. It is claimed that 
such track is better for heavy and fast traffic, but it is more 




Fig. 112. — Bull- 
headed Rail and 
Chair. 



258 



RAILROAD CONSTRUCTION. 



§ 226. 



expensive to build and maintain. It is the standard form of 
track in England and some parts of Europe. 

Until a few years ago there was a ver}^ great multiplicity 
in the designs of ^'T" rails as used in this country, nearly every 
prominent railroad having its own special design, Avhich perhaps 
differed from that of some other road by only a very minute and 
insignificant detail, but which nevertheless would require a 
complete new set of rolls for rolling. This certainly must have 
had a very appreciable effect on the cost of rails. In 1893, the 
American Society of Civil Engineers, after a very exhaustive 
investigation of the subject, extending over several years, hav- 
ing obtained the opinions of the best experts of the country, 
adopted a series of sections which have been very extensively 
adopted by the railroads of this country. Instead of having 
the rail sections for various weights to be geometrically similar 
figures, certain dimensions are made constant, regardless of the 
weight. It was decided that the metal should be distributed 
through the section in the proportions of — head 42%, web 21%, 
and flange 37%. The top of the head should have a radius of 




Fig. 113. — Am. Soc. C. E. Standard Rail Section. 



12"; the top corner radius of head should be y%"; the lower 
corner radius of head should be y^^"; the corners of the flanges, 
yV radius; side radius of web, 12"; top and bottom radii of 
web corners, J"; and angles with the horizontal of the under side 



§227. 



RAILS. 



259 



of the head and the top of the flange, 13°. The sides of the head 
are vertical. 

The height of the rail (D) and the width of the base (C) are 
always made equal to each other. 





Weight per Yard. 




40 


45 


50 


55 


60 


65 


70 


75 


80 


85 


90 


95 


100 


A 


IF 


2" 


2V 


2V 


2r 


2W 


2iV' 


2^r 


2r 


2iV' 


2r 


2ir 


21" 


B 


II 


11 


T^ 


\l 


li 


i 


II 


§1 


35 


r% 


i*« 


T% 


^% 


C &D 


3^ 


m 


3i 


4t^5 


4i 


4/b 


41 


411 


5 


5,^ 


5f 


Sii 


51 


E 


i 


u 


\l 


§1 


II 


if 


il 


1^ 


i 


il 


il 


ii 


§i 


F 


HI 


m 


2^ 


2H 


211 


21 


2hl 


211 


2i 


2i 


211 


211 


3/* 


G 


Is^? 


ii^ 


n 


Ui 


U^ 


Ih 


ni 


HI 


U 


HI 


Ul 


HI 


HI 



The chief features of disagreement among railroad men relate 
to the radius of the upper corner of the head and the slope of the 
side of the head. The radius (/g") adopted for the upper corner 
(constant for all weights) is a little more than is advocated by 
those in favor of ''sharp corners" who often use a radius of i". 
On the other hand it is much less than is advocated by those 
who consider that it should be nearly equal 
to (or even greater than) the larger radius 
universally adopted for the corner of the 
wheel-flange. The discussion turns on the 
relative rapidity of rail wear and the wear 
of the wheel-flanges as affected by the rela- 
tion of the form of the wheel-tread to that 
of the rail. It is argued that sharp rail 
corners wear the wheel-flanges so as to 
produce sharp flanges, which are liable to 

FiQ. 114. — Relation cause derailment at switches and also to 
OF Rail TO Wheeij- . .i i xv x* c • j • 

TREAD. require that the tires of engine-drivers 

must be more frequently turned down to their true form. On 

the other hand it is generally believed that rail wear is much less 

rapid while the area of contact between the rail and wheel-flange 

is small, and that when the rail has worn down, as it invariably 

does, to nearly the same form as the wheel-flange, the rail wears 

away very quickly. 

227. Weight for various kinds of traffic. The heaviest rails 

in regular use weigh 100 lbs. per yard, and even these are only 

used on some of the heaviest traffic sections of such roads as the 




260 RAILROAD CONSTRUCTION. § 227. 

N. Y. Central, the Pennsylvania, the N. Y., N. H. & H., and 
a few others. Probably the larger part of the mileage of the 
country is laid with 60- to 75-lb. rails — considering the fact that 
^'the larger part of the mileage" consists of comparatively hght- 
traffic roads and may exclude all the heavy trunk lines. Very 
light-traffic roads are sometimes laid with 56-lb. rails. Roads 
with fairly heavy traffic generally use 75- to 85-lb. rails, espe- 
cially when grades are heavy and there is much and sharp cuj-va- 
ture. The tendency on all roads is toward an increase in the 
weight, rendered necessary on account of the increase in the 
weight and capacity of rolling stock, and due also to the fact that 
the price of rails has been so reduced that it is both better and 
cheaper to obtain a more solid and durable track b}^ increasing 
the weight of the rail rather than by attempting to support a 
weak rail by an excessive number of ties or by excessive track 
labor in tamping. It should be remembered that in buying rails 
the mere weight is, in one sense, of no importance. The im- 
portant thing to consider is the strength and the stiffness. If 
we assume that all weights of rails have similar cross-sections 
(which is nearly although not exactly true), then, since for beams 
of similar cross-sections the strength varies as the cube of the 
homologous dimensions and the stiffness as the fourth power ^ 
while the area (and therefore the weight per unit of length) 
only varies as the square, it follows that the stiffness varies as 
the square of the weight, and the strength as the f power of the 
weight. Since for ordinary variations of weight the price per 
ton is the same, adding (say) 10% to the weight (and cost) adds 
^1% to the stiffness and over 15% to the strength. As another 
illustration, using an 80-lb. rail instead of a 75-lb. rail adds only 
6|% to the cost, but adds about 14% to the stiffness and nearly 
11% to the strength. This shows why heavier rails are more 
economical and are being adopted even when they are not abso- 
lutely needed on account of heavier rolling stock. The stiffness, 
strength, and consequent durability are increased in a much 
greater ratio than the cost. 

228. Effect of stiffness on traction. A very important but 
generally unconsidered feature of a stiff rail is its effect on trac- 
tive force. An extreme illustration of this principle is seen 
when a vehicle is drawn over a soft sandy road. The constant 
compression of the sand in front of the wheel has virtually the 
same effect on traction as drawing the wheel up a grade whose 



§ 229. RAILS. 261 

steepness depends on the radius of the wheel and the depth of 
the rut. On the other hand, if a wheel, made of perfectly 
elastic material, is rolled over a surface which, while supported 
with absolute rigidity, is also perfectly elastic, there would be a 
forward component, caused by the expanding of the compressed 
metal just behind the center of contact, which would just bal- 
ance the backw^ard component. If the rail was supported 
throughout its length by an absolutely rigid support, the high 
elasticity of the wheel-tires and rails would reduce this form of 
resistance to an insignificant quantity, but the ballast and even 
the ties are comparatively inelastic. When a weak rail yields, 
the ballast is more or less compressed or displaced, and even 
though the elasticity of the rail brings it back to nearly its 
former place, the work done in compressing an inelastic material 
is wholly lost. The effect of this on the fuel account is certainly 
very considerable and yet is frequently entirely overlooked. It 
is practically impossible to compute the saving in tractive power, 
and therefore in cost of fuel, resulting from a given increase in 
the weight and stiffness of the rail, since the yielding of the rail 
is so dependent on the spacing of the ties, the tamping, etc. But 
it is not difficult to perceive in a general way that such an econ- 
omy is possible and that it should not be neglected in considering 
the value of stiffness in rails. 

229. Length of rails. The recommended standard minimum 
length of rails is 33 feet. In recent years many roads have been 
trying 45-foot and even 60-foot rails. The argument in favor of 
longer rails is chiefly that of the reduction in track-joints, which 
are costly to construct and to maintain and are a fruitful source 
of accidents. Mr. Morrison of the Lehigh Valley R. R.* declares 
that, as a result of extensive experience with 45-foot rails on 
that road, he finds that they are much less expensive to handle, 
and that, being so long, they can be laid around sharp curves 
without being curved in a machine, as is necessary with the 
shorter rails. The great objection to longer rails lies in the 
difficulty in allowing for the expansion, which will require, in 
the coldest weather, an opening at the joint of nearly f" for a 
60-foot rail. The Pennsylvania R. R. and the Norfolk and 
Western R. R. each have a considerable mileage laid with 60-foot 
rails. 

* Report, Roadmasters Association, 1895. 



262 RAILROAD CONSTRUCTION. § 230. 

230. Expansion of rails. Steel expands at the rate of .0000065 
of its length per degree Fahrenheit. The extreme range of tem- 
perature to which any rail will be subjected will be about 160°, 
or say from -20° F. to +140° F. With the above coefficient 
and a rail length of 60 feet the expansion would be 0.0624 foot, 
or about J inch. But it is doubtful whether there would ever 
be such a range of motion even if there were such a range of 
temperature. Mr. A. Torrey, chief engineer of the Mich. Cent. 
R. R., experimented with a section over 500 feet long, which, 
although not a single rail, was made ^' continuous '* by rigid 
splicing, and he found that there was no appreciable additional 
contraction of the rail at any temperature below +20° F. The 
reason is not clear, but the fact is undeniable. 

The heavy girder rails, used by the street railroads of the 
country, are bonded together with perfectly tight rigid joints 
which do not permit expansion. If the rails are laid at a tem- 
perature of 6t° F. and the temperature sinks to 0°, the rails 
have a tendency to contract .00039 of their length. If this 
tendency is resisted by the friction of the pavement in which the 
rails are buried, it only results in a tension amounting to .00039 
of the modulus of elasticity, or say 10920 pounds per square 
inch, assuming 28 000 000 as the modulus of elasticity. This 
stress is not dangerous and may be permitted. If the tempera- 
ture rises to 120° ¥., a tendency to expansion and buckling will 
take place, which will be resisted as before by the pavement, 
and a compression of 10920 pounds per square inch will be in- 
duced, which will likewise be harmless. The range of tempera- 
ture of rails which are buried in pavement is much less than 
when they are entirely above the ground and wiU probably 
never reach the above extremes. Rails supported on ties which 
are only held in place by ballast must be allowed to expand and 
contract almost freely, as the ballast cannot be depended on to 
resist the distortion induced by an}^ considerable range of tem- 
perature, especially on curves. 

231. Rules for allowing for temperature. Track regulations 
generally require that the track foremen shall use iron (not 
wooden) shims for placing between the ends of the rails while 
splicing them. The thickness of these shims should vary with 
the temperature. Some roads use such approximate rules as the 
following: ^'The proper thickness for coldest Aveather is j^ of an 
inch; during spring and fall use J of an inch, and in the very 



§ 232. 



RAILS. 



263 



hottest weather re of an inch should be allowed." This is on 
the basis of a 30-foot rail. When a more accurate adjustment 
than this is desired, it may be done by assuming some very high 
temperature (100° to 125° F.) as a maximum, when the joints 
should be tight; then compute in tabular form the spacing for 
each temperature, varying by 25°, allowing 0".0643 (very 
nearly re") for each 25° change. Such a tabular form would 
be about as follows (rail length 33 feet): 



Temperature . . 


Over 100° 


100°-75° 


75°-50° 


50°-25° 


25°-0° 


Below 0° 


Rail opening . . . 


Close 


^" 


V 


A" 


1." 

4 


^"" 



One practical difficulty in the way of great refinement in this 
work is the determination of the real temperature of the rail 
when it is laid. A rail lying in the hot sun has a very much 
higher temperature than the air. The temperature of the rail 
cannot be obtained even by exposing a thermometer directly to 
the sun, although such a result might be the best that is easily 
obtainable. On a cloudy or rainy day the rail has practically 
the same temperature as the air; therefore on such days there 
need be no such trouble. 

232. Chemical composition. About 98 to 99.5% of the com- 
position of steel rails is iron, but the value of the rail, as a rail, 
is almost wholly dependent upon the large number of other 
chemical elements which are, or may be, present in very small 
amounts. The manager of a steel-rail mill once declared that 
their aim was to produce rails having in them — 

Carbon 0.32 to 0.40% ' 

Silicon 0.01 to 0.06% 

Phosphorus 0.09 to 0. 105% 

Manganese 1 . 00 to 1 . 50% 

The analysis of 32 specimens of rails on the Chic, Mil. & St. 
Paul R. R. showed variations as follows: 



Carbon 0.211 to 0.52% 

SiHcon. , 0.013 to 0.256% 

Phosphorus 0.055 to 0. 181% 

Manganese. . 0.35 to 1 .63% 



264 RAILROAD CONSTRUCTION. § 232. 

These quantities have the same general relative proportions 
as the rail-mill standard given above, the differences lying 
mainly in the broadening of the limits. Increasing the per- 
centage of carbon by even a few hundredths of one per cent 
makes the rail harder, but likewise more brittle. If a track is 
well ballasted and not subject to heaving by frost, a harder and 
more brittle rail may be used without excessive danger of break- 
age, and such a rail will wear much longer than a softer tougher 
rail, although the softer tougher rail may be the better rail for 
a road having a less perfect roadbed. 

A small but objectionable percentage of sulphur is some- 
times found in rails, and very delicate analysis will often show 
the presence, in very minute quantities, of several other chem- 
ical elements. The use of a very small quantity of nickel or 
aluminum has often been suggested as a means of producing 
a more durable rail. The added cost and the uncertainty of 
the amount of advantage to be gained has hitherto prevented 
the practical use or manufacture of such rails. 

233. Testing. Chemical and mechanical testing are both 
necessary for a thorough determination of the value of a rail. 
The chemical testing has for its main object the determination 
of those minute quantities of chemical elements which have such 
a marked influence on the rail for good or bad. The mechanical 
testing consists of the usual tests for elastic limit, ultimate 
strength, and elongation at rupture, determined from pieces cut 
out of the rail, besides a ''drop test." The drop test consists 
in dropping a weight of 2000 lbs. from a height of 18 to 22 feet 
on to the center of a rail which is supported on abutments, 
placed three feet apart. The number of blows required to 
produce rupture or to produce a permanent set of specified 
magnitude gives a measure of the strength and toughness of 
the rail. 

233a. Proposed standard specifications for steel rails. The 
following specifications for steel rails are those proposed by a 
committee of the American Railway Engineering and Main- 
tenance of Way Association in March, 1902: 

1. (a) Steel may be made by the Bessemer or open-hearth 
process. 

(b) The entire process of manufacture and testing shall be in 
accordance with the best standard current practice, and special 
care shall be taken to conform to the folloT\4ng instructions: 



§ 233. 



RAILS. 



265 



(c) Ingots shall be kept in a vertical position in pit-heating 
furnaces. 

(d) No bled ingots shall be used. 

(e) Sufficient material shall be discarded from the top of the 
ingots to insure sound rails. 



CHEMICAL PROPERTIES. 



2. Rails of the various weights per yard specified below shall 
conform to the following limits in chemical composition: 



Carbon 

Phosphorus shall not 
exceed 

Silicon shall not ex- 
ceed 

Manganese 



50 to 59 + 

Ibs. 
per cent. 



0.35-0.45 
0.10 



0.20 
0.70-1.00 



60 to 69 + 

lbs. 
per cent. 



0.38-0.48 
0.10 



0.20 
0.70-1.00 



70 to 79 + 

lbs. 
per cent. 



0.40-0.50 

0.10 

0.20 
0.75-1.05 



80 to 89 + 

lbs. 
per cent. 



0.43-0.53 

0.10 

0.20 
0.80-1.10 



90 to 100 

lbs. 
per cent. 



0.45-0.55 

0.10 

0.20 
0.80-1.10 



PHYSICAL PROPERTIES. 

3. One drop test shall be made on a piece of rail not more than 
6 feet long, selected from every fifth blow of steel. The test- 
piece shall be taken from the top of the ingot. The rail shall 
be placed head upwards on the supports and the various sections 
shall be subjected to the following impact tests: 



Weight of Rail in Pounds per Yard. 



45 to and including 55 

More than 55 " " *' 65 

" 65 " " " 75 

" 75 " •' " 85 

'' 85 " •• '* 100 



Height of Drop 
in Feet. 



15 
16 
17 
18 
19 



If any rail break when subjected to the drop test two additional 
tests will be made of other rails from the same blow of steel, and 
if either of these latter tests fail, all the rails of the blow which 
they represent will be rejected; but if both of these additional 
test-pieces meet the requirements all the rails of the blow which 
they represent will be accepted. If the rails from the tested 
blow shall be rejected for failure to meet the requirements of 



266 RAILROAD CONSTRUCTION. § 233. 

the drop test, as above specified, two other rails will be subjected 
to the same tests, one from the blow next preceding and one from 
the blow next succeeding, the rejected blow. In case the first 
test taken from the preceding or succeeding blow shall fail two 
additional tests shall be taken from the same blow of steel, the 
acceptance or rejection of which shall also be determined as 
specified above, and if the rails of the preceding or succeeding, 
blow shall be rejected, similar tests may be taken from the pre- 
vious or following blows, as the case may be, until the entire 
group of five blows is tested, if necessary. The acceptance or 
rejection of all rails from any blow will depend upon the results 
of the tests thereof. 

HEAT TREATMENT. 

The number of passes and speed of train shall be so regulated 
that on leaving the rolls at the final pass the temperature of the 
rail will not exceed that which requires a shrinkage allowance at 
the hot saws of 6 inches for 85-lb. and 6J inches for 100-lb. rails, 
and no artificial means of cooling the rails shall be used between 
the finishing pass and the hot saws. 

TEST-PIECES AND METHODS OF TESTING. 

4. The drop-test machine shall have a tup of 2000 lbs. weight, 
the striking face of which shall have a radius of not more than 
5 inches, and the test rail shall be placed head upwards on solid 
supports 3 feet apart. The anvil-block shall weigh at least 
20000 lbs., and the support shall be a part of, or firmly secured 
to, the anvil. 

. 5. The manufacturer shall furnish the inspector, daily, mth 
carbon determinations of each blow, and a complete chemical 
analysis every 24 hours, representing the average of the other 
elements contained in the steel. These analyses shall be made 
on drillings taken from a small test ingot. 

FINISH. 

6. Unless otherwise specified the section of rail shall be the 
American standard, recommended by the American Society of 
Civil Engineers, and shall conform, as accurately as possible, 
to the templet furnished by the railroad company, consistent 
with paragraph No. 7, relative to the specified weight. A vari- 



§ 233. RAILS. 267 

ation in height of ^^^ inch less and 3V inch greater than the specified 
height will be permitted. A perfect fit of the splice-bars, how- 
ever, shall be maintained at all times. 

7. The weight of the rails shall be maintained as nearly as 
possible, after complying with paragraph No. 6, to that specified 
in contract. A variation of one-half of one per cent for an entire 
order will be allowed. Rails shall be accepted and paid for ac- 
cording to actual weights. 

8. The standard length of rails shall be 33 feet. Ten per cent 
of the entire order will be accepted in shorter lengths, varying 
by even feet down to 27 feet. A variation of \ inch in length 
from that specified will be allowed. 

9. Circular holes for splice-bars shall be drilled in accordance 
with the specifications of the purchaser. The holes shall accu- 
rately conform to the drawing and dimensions furnished in every 
respect, and must be free from burrs. 

10. Rails shall be straightened while cold, smooth on head, 
sawed square at ends, and, prior to shipment, shall have the 
burr, occasioned by the saw-cutting, removed, and the ends 
made clean. No. 1 rails shall be free from injurious defects and 
flaws of all kinds. 

BRANDING. 

11. The name of the maker, the month and year of manu- 
facture shall be rolled in raised letters on the side of the web, 
and the number of the blow shall be stamped on each rail 

INSPECTION. 

12. The inspector representing the purchaser shall have all 
reasonable facihties afforded to him by the manufacturer to 
satisfy him that the finished material is furnished in accord- 
ance with these specifications. All tests and inspections shall 
be made at the place of manufacture, prior to shipment. 

NO. 2 RAILS. 

13. Rails that possess any injurious physical defects, or which 
for any other cause are not suitable for first quality, or No. 1 
rails, shall be considered as No. 2 rails, provided, however, that 
rails which contain any physical defects which seriously impair 
their strength shall be rejected. The ends of all No. 2 rails 
shall be painted in order to distinguish them. 



268 



RAILROAD CONSTRUCTION. 



§234. 




Fig. 115. 



234. Rail wear on tangents. When the wheel loads on a rail 
are abnormally heavy, and particularly when the rail has but 
little carbon and is unusually soft, the concentrated pressure 
on the rail is frequently greater than the 

elastic limit, and the metal ''flows'' so that 
the head, although greatly abraded, will 
spread somewhat outside of its original lines, 
as shown in Fig. 115. The rail wear that 
occurs on tangents is almost exclusively 
on top. Statistics show that the rate of 
rail wear on tangents decreases as the rails 
are more worn. Tests of a large number of 
rails on tangents have shown a rail wear averaging nearly one 
pound per yard per 10 000 000 tons of traffic. There is about 
33 pounds of metal in one yard of the head of an 80-lb. rail. As 
an extreme value this may be worn down one-haLf, thus giving 
a tonnage of 165 000 000 tons for the life of the rail. Other 
estimates bring the tonnage down to 125 000 000 tons. Since 
the locomotive is considered to be responsible for one-half (and 
possibly more) of the damage done to the rail, it is found that 
the rate of wear on roads with shorter trains is more rapid in 
proportion to the tonnage, and it is therefore thought that the 
life of a rail should be expressed in terms of the number of trains. 
This has been estimated at 300 000 to 500 000 trains. 

235. Rail wear on curves. On curves the maximum rail wear 
occurs on the inner side of the head of the outer rail, giving a 
worn form somewhat as shown in Fig. 116. The dotted line 

shows the nature and progress of the rail wear 
on the inner rail of a curve. Since the press- 
ure on the outer rail is somewhat lateral 
rather than vertical, the ''flow" does not 
take place to the same extent, if at all, on 
the outside, and whatever flow would take 
place on the inside is immediately worn off 
by the wheel-flange. Unlike the wear on 
tangents, the wear on curves is at a greater 
rate as the rail becomes more worn. 

The inside rail on curves wears chiefly on top, the same as 
on a tangent, except that the wear is much greater owing to the 
longitudinal slipping of the wheels on the rail, and the lateral 
slipping that must occur when a rigid four-wheeled truck is 




Fig. 116. 



§ 236. RAILS. 269 

guided around a curve. The outside rail is subjected to a 
greater or less proportion of the longitudinal slipping, likewise 
to the lateral slipping^ and, worst of all, to the grinding action 
of the flange of the wheel, which grinds off the side of the head. 

The results of some very elaborate tests, made by Mr. A M. 
Wellington, on the Atlantic and Great Western R. R., on the 
wear of rails, seem to show that the rail wear on curves may be 
expressed by the formula: ''Total wear of rails on a d degree 
curve in pounds per yard per 10 000 000 tons duty = 1 4-0.03(i^.^' 
"It is not pretended that this formula is strictly correct even 
in theory, but several theoretical considerations indicate that 
it may hh nearly so." According to this formula the average 
rail wear on a 6° curve will be about twice the rail wear on a tan- 
gent. While this is approximately true^ the various causes 
modifying the rate of rail wear (length of trains^ age and quality 
of rails, etc.) will result in numerous and large A^ariations from 
the above formula, which should only be taken as indicating an 
approximate law. 

236. Cost of rails. In 1873 the cost of steel rails was about 
$120 per ton, and the cost of iron rails about S70 per ton 
Although the steel rails were at once recognized as superior to 
iron rails on account of more uniform Avear, they were an expen- 
sive luxury. The manufacture of steel rails by the Bessemer 
process created a revolution in prices, and they steadily dropped 
in price until, several years ago, steel rails were manufactured 
and sold for $22 per ton. For several years past the price has 
been very uniform at S28 per ton at the mill. At such prices 
there is no longer any demand for iron rails, since the cost of 
manufacturing them is substantially the same as that of steel 
rails, while their durability is unquestionably inferior to that of 
steel rails. 



CHAPTER X. 

RAIL-FASTENINGS. 

RAIL-JOINTS 

237. Theoretical requirements for a perfect joint. A perfect 
rail-joint is one that has the same strength and stiffness — no 
more and no less — as the rails which it joins, and which will 
not interfere with the regular and uniform spacing of ties. It 
should also be reasonably cheap both in first cost and in cost of 
maintenance. Since the action of heavy loads on an elastic rail 
is to cause a wave of translation in front of each wheel, any 
change in the stiffness or elasticity of the rail structure will 
cause more or less of a shock, which must be taken up and 
resisted by the joint. The greater the change in stiffness the 
greater the shock, and the greater the destructive action of the 
shock. The perfect rail-joint must keep both rail-ends truly in 
line both laterally and vertically, so that the flange or tread of 
the wheel need not jump or change its direction of motion sud- 
denly in passing from one rail to the other. A consideration of 
all the above requirements will show that only a perfect welding 
of rail-ends would produce a joint of uniform strength and stiff- 
ness which would give a uniform elastic wave ahead of each 
wheel. As welding is impracticable for ordinary railroad work 
(see § 230), some other contrivance is necessary which will 
approach this ideal as closely as may be. 

238. Efficiency of the ordinary angle-bar. Throughout the 
middle portion of a rail the rail acts as a continuous girder. If 
we consider for simplicity that the ties are unyielding, the deflec- 
tion of such a continuous girder between the ties will be but 
one-fourth of the deflection that would be found if the rail were 
cut half-way between the ties and an equal concentrated load 
were divided equally between the two unconnected ends. The 
maximum stress for the continuous girder would be but one-half 
of that in the cantilevers. Joining these ends with rail-joints 
will give the ordinary ^'suspended" joint. In order to main- 

270 



§ 239. RAIL-FASTENINGS. 271 

tain uniform strength and stiffness the angle-bars must supply 
the deficiency. These theoretical relations are modified to an 
unknown extent by the unknown and variable jaelding of the 
ties From sonie experiments made by the Association of 
Engineers of Maintenance of Way of the P. R. R.* the following 
deductions were made* 

1. The capacity of a ^'suspended" joint is greater than that 
of a "supported" joint — whether supported on one or three 
ties. (See §240) 

2. That (with the particular patterns tested) the angle-bars 
alone can carry only 53 to 56% of a concentrated load placed 
on a joint. 

3. That the capacity of the w^hole joint (angle-bars and rail) 
is only 52 4% of the strength of the unbroken rail. 

4. That the ineffectiveness of the angle-bar is due chiefly to 
a deficiency in compressive resistance. 

Although it has been universally recognized that the angle- 
bar is not a perfect form of joint, its simplicity, cheapness, and 
reliability have caused its almost universal adoption. Within a 
very few years other forms (to be described later) have been 
adopted on trial sections and have been more and more extended, 
until their present use is very large. The present time (1900) is 
evidently a transition period, and it is quite probable that within 
a very few years the now common angle -plate will be as un- 
known in standard practice as the old-fashioned "fish-plate" 
is at the present time. 

239. Effect of rail gap at joints. It has been found that the 
jar at a joint is due almost entirely to the deflection of the joint 
and scarcely at all to the small gap required for expansion. 
This gap causes a drop equal to the versed sine of the arc having 
a chord equal to the gap and a radius equal to the radius of 
the wheel. Taking the extreme case (for a 30-foot rail) of a I" 
gap and a 33" freight-car wheel, the drop is about joVu''- In 
order to test how much the jarring at a joint is due to a gap be- 
tween the rails, the experiment was tried of cutting shallow 
notches in the top of an otherwise solid rail and running a loco- 
motive and an inspection car over them. The resulting jarring 
was practically imperceptible and not comparable to the jar pro- 
duced at joints. Notwithstanding this fact, many plans have 

* Roadmasters Association of America — Reports for 1897. 



272 RAILROAD CONSTRUCTION. § 239. 

been tried for avoiding this gap. The most of these plans con- 
sist essentially of some form of compound rail, the sections 
breaking joints. (Of course the design of the compound rail 
has also several other objects in view.) In Fig. 117 are shown a 




Fig. 117. — Compound Rail Sections. 

few of the very many designs which have been proposed. These 
designs have invariably been abandoned after trial. Another 
plan, which has been extensively tried on the Lehigh Valley 
R. R., is the use of mitered joints. The advantages gained by 
their use are as yet doubtful, while the added expense is unques- 
tionable. The ^' Roadmasters Association of America" in 1895 
adopted a resolution recommending mitered joints for double 
track, but their use does not seem to be growing. 

240. ** Supported," " suspended," and " bridge " joints. In a 
supported joint the ends of the rails are on a tie. If the angle- 
plates are short, the joint is entirely supported on one tie; if 
very long, it may be possible to place three ties under one angle- 
bar and thus the joint is virtually supported on three ties rather 
than one. In a suspended joint the ends of the rails are midway 
between two ties and the joint is supported by the two. There 
have always been advocates of both methods, but suspended 
joints are more generally used than supported joints. The 
opponents of three-tie joints claim that either the middle tie will 
be too strongly tamped, thus making it a supported joint, or 
that, if the middle tie is weakest, the joint becomes a very long 
(and therefore weak) suspended joint between the outer joint- 
ties, or that possibly one of the outer joint-ties gives way, thus 
breaking the angle-plate at the joint. Another objection which 
is urged is that unless the bars are very long (say 44 inches, as 
used on the Mich. Cent. R. R.) the ties are too close for proper 
tamping. The best answer to these objections is the successful 
use of these joints on several heavy-traffic roads 

''Bridget-joints are similar to suspended joints in that the 
joint is supported on two ties, but there is the important differ- 
ence that the bridge joint supports the rail from underneath and 



§ 242. RAIL-FASTENINGS. 273 

there is no transverse stress in the rail, whereas the suspended 
joint requires the combined transverse strength of both angle- 
bars and rail. A serious objection to bridge-joints lies in the 
fact of their considerable thickness between the rail base and the 
tie. When joints are placed ^^ staggered'^ rather than ''oppo- 
site*' (as is now the invariable standard practice), the ties sup- 
porting a bridge-joint must either be notched down, thus weak- 
ening the tie and promoting decay at the cut, or else the tie 
must be laid on a slope and the joint and the opposite rail do not 
get a fair bearing. 

241. Failures of rail-joints. It has been observed on double- 
track roads that the maximum rail wear occurs a few inches 
beyond the rail gap at the joint in the direction of the traffic. 
On single-track roads the maximum rail wear is found a few 
inches each side of the joint rather than at the extreme ends of 
the rail, thus showing that the rail end deflects down under the 
wheel until (with fast trains especially) the wheel actually jumps 
the space and strikes the rail a few inches beyond the joint, the 
impact producing excessive wear. This action, which is called 
the ''drop," is apt to cause the first tie beyond the joint to 
become depressed, and unless this tie is carefully watched and 
maintained at its proper level, the stresses in the angle-bar may 
actually become reversed and the bar may break at the top. The 
angle-bars of a suspended joint are normally in compression at 
the top. The mere reversal of the stresses would cause the bars 



Fig. 118. — Effect of "Wheel Drop" (Exaggerated). 

to give way with a less stress than if the stress were always the 
same in kind. A supported joint, and especially a three-tie 
joint (see § 240), is apt to be broken in the same manner. 

242. Standard angle- bars. An angle-bar must be so made 
as to closely fit the rails. The great multiplicity in the designs 
of rails (referred to in Chapter IX) results in nearly as great 
variety in the detailed dimensions of the angle-bars. The sec- 
tions here illustrated must be considered only as types of the 
variable forms necessary for each different shape of rail. The 



274 



RAILROAD CONSTRUCTION. 



? 242. 



S^ 



absolutely essential features required for a fit are (1) the angles 
of the upper and lower surfaces of the bar where they fit against 
the rail, and (2) the height of the bar. The bolt-holes in the 
bar and rail must also correspond. The holes in the angle-plates 
are elongated or made oval, so that the track-bolts, which are 




|«Hrf*i*H i^'A- *\' i^Vi^ *i^lf—o%- J 

Fig. 119.— Standard Angle-bar— 80-lb. Rail. M. C. R.R. 

made of corresponding shape immediately under the head, will 
not be turned by jarring or \^bration. The holes in the rails 
are made of larger diameter (by about A") than the bolts, so as 
to allow the rail to expand with temperature. 

The standard drilling for bolt-holes in the end of a rail, as 
adopted by the A. R. E. ik M. W. Assoc, in 1906, is as follows: 

End of rail to center first hole 2ii ins. 

Center first hole to center second hole 5 

Center second hole to center third hole (for six-hole 

joint) ^ 

The proper length of angle-bars has not yet been standard- 
ized, but the above dimensions point very closely to the proper 
length. If the centers of the middle pair of holes in the angle- 
plate are made 4J inches apart, and the holes in the rails are 
A inch larger in diameter than the bolts, it will allow for an 
extreme variation in the length of the rails of | inch— due to 
expansion. Adding 4 inches at each end of the joint, from 



§ 243. RAIL-FASTENINGS. 275 

the center of the last hole to the end of the angle-plate, will 
make a length of 22| inches for a four-hole and 32f inches for 
a six-hole joint. This is considerably less than the M. C. R. R. 
joint shown above, but this joint was purposely lengthened so 
that it could be used for a three-tie joint. 

243. Later designs of rail-joints. In Plate VII are shown 
various designs which are competing for adoption. The most 
prominent of these (judging from the discussion in the conven- 
tion of the Roadmasters Association of America in 1897) are 
the 'Continuous" and the ''Weber." Each of them has been 
very extensively adopted, and where used are universally pre- 
ferred to angle-plates. Nearly all the later designs embody 
more or less directly the principle of the bridge-joint, i.e., sup- 
port the rail from underneath. An experience of several years 
will be required to demonstrate which form of joint best satis- 
fies the somewhat opposed requirements of minimum cost (both 
initial and for maintenance) and minimum wear of rails and 
rolling stock. 

243a. Proposed specifications for steel splice-bars. The fol- 
lowing specifications for steel splice-bars were proposed in 1900 
by Committee No. 1, American Section, International Associa- 
tion for Testing Materials. 

1. Steel for splice-bars may be made by the Bessemer or open- 
hearth process. 

2. Steel for sphce-bars shall conform to the following limits 
in chemical composition: 

Per cent. 

Carbon shall not exceed 0.15 

Phosphorus shall not exceed 0.10 

Manganese 0.30 to 0.60 

3. Splice-bar steel shall conform to the following physical 
qualities : 

Tensile strength, pounds per square inch 54000 to 64000 

Yield point, pounds per square inch 32000 

Elongation, per cent in eight inches shall not 

be less than 25 

4. (a) A test specimen cut from the head of the spHce-bar 
shall bend 180° flat on itself without fracture on the outside 
of the bent portion. 



276 RAILROAD CONSTRUCTION. § 243. 

(h) If preferred the bending test may be made on an un- 
punched splice-bar, which, if necessary, shall be first flattened 
and shall then be bent 180° flat on itself without fracture on 
the outside of the bent portion. 

5. A test specimen of 8-inch gauged length, cut from the head 
of the splice-bar, shall be used to determine the physical proper- 
ties specified in paragraph No. 3. 

6. One tensile specimen shall be taken from the rolled splice- 
bars of each blow or melt, but in case this develops flaws, or 
breaks outside of the middle third of its gauged length, it may 
be discarded and another test specimen submitted therefor. 

7. One test specimen cut from the head of the splice-bar shall 
be taken from a rolled bar of each blow or melt, or if preferred 
the bending test may be made on an unpunched splice-bar, 
which, if necessary, shall be flattened before testing. The bend- 
ing test may be made by pressure or by blows. 

8. For the purposes of this specification, the yield point shall 
be determined by the careful observation of the drop of the 
beam or halt in the gauge of the testing machine. 

9. In order to determine if the material conforms to the chem- 
ical limitations prescribed in paragraph No. 2 herein, analysis 
shall be made of drillings taken from a small test ingot. 

10. All splice-bars shall be smoothly rolled and true to templet. 
The bars shall be sheared accuratel}^ to length and free from 
fins or cracks, and shall perfectly fit the rails for which they are 
intended. The punching and notching shall accurately conform 
in every respect to the drawing and dimensions furnished. 

11. The name of the maker and the year of manufacture shall 
be rolled in raised letters on the side of the splice-bar. 

12. The inspector representing the purchaser shall have all 
reasonable facilities afforded to him by the manufacturer, to 
satisfy him that the finished material is furnished in accordance 
with these specifications. All tests and inspections shall be 
made at the place of manufacture, prior to shipment. 

TIE-PLATES. 

244. Advantages, (a) As already indicated in § 204, the 
life of a soft-wood tie is very much reduced by " rail-cutting '* 
and '' spike-killing,'' such ties frequently requiring renewal long 




CLOUD JOINT 



t 



'ZiJ 




ATLAS SUSPENDED RAIL JOINT. 



FISHER BRIDGE JOINT. 




WOLHAUPTER JOINT 




WEBER RAIL JOINT. 



Plate VII. — Some forms of Rail Joints. 
{Between pp. 275 and 276.) 




ZAND RAIL JOINT. 



r^ 



§ 244. RAIL-FASTENINGS. 277 

before any serious decay has set in. It has been practically 
demonstrated that the ^'rail-cutting'^ is not due to the mere 
pressure of the rail on the tie, even with a maximum load on 
the rail, but is due to the impact resulting from vibration and 
to the longitudinal working of the rail. It has been proved 
that this rail-cutting is jjractically prevented by the use of tie- 
plates, (b) On curves there is a tendency to overturn the outer 
rail due to the lateral pressure on the side of the head. 
This produces a concentrated pressure of the outer edge of the 
base on the tie which produces rail-cutting and also draws the 
inner spikes. Formerly the only method of guarding against 
this was by the use of '^ rail-braces/' one pattern of which is 
showTi in Fig. 120. But it has been found that tie-plates serve 
the purpose even better, and rail-braces have been abandoned 
where tie-plates are used, (c) Driving spikes through holes 
in the plate enables the spikes on each side of the rail to mutually 
support each other, no matter in which (lateral) direction the 
rail may tend to move, and this probably accounts in large 




PFio. 120. — Atlas Brace K; 

measure for the added stability obtained by the use of tie-plates. 
id) The wear in spikes, called ^^ necking,'' caused by the ver- 
tical vibration of the rail against them, is very greatly reduced. 
(e) The cost is very small compared with the value of the added 
life of the tie, the large reduction in the work of track main- 
tenance, and the smoother running on the better track which is 
obtained. It has been estimated that by the use of tie-plates 
the life of hard-wood ties is increased from one to three years, 
and the life of soft-wood ties is increased from three to six 
years. From the very nature of the case, the value of tie-plates 
is greater when they are used to protect soft ties. 



278 



RAILROAD CONSTRUCTION. 



§ 245. 



245. Elements of the design. There is still a great diversity 
of opinion regarding the relative advantages of tie-plates which 
are flat on the bottom and those which are corrugated or pro- 
vided with teeth or claws which are imbedded in the tie. Those 
used in Europe are without exception flat on the bottom. The 
Pennsylvania Railroad and the Southern Pacific have also 
been using flat tie-plates. On the other hand, it is claimed 
that the pressure required to force a corrugated plate into a 
tie is about 20% greater than that required to imbed a flat 




Wolhaupter 





Round grooved-tapered-flat 
bottom-shoulder tie plate 



^ 



P.R.R. flat bottom tie plate 
Claw and shoulder tie plate 

Fig. 121. — Various Forms of Tie-plates. 



plate of equal thickness in the tie. It is also claimed that tests 
have shown that the force required to spread the rails when 
they are fastened with corrugated plates under the rails is 
from 36% to 100% greater than that required when a flat 
tie-plate is used. It is especially important that the plate 
shall be so firmly imbedded in the tie that it cannot move or 
''rock'' with each motion of the rail over it. Instances are 
known where a treated tie has become unfit for service because 



§ 245. RAIL-FASTENINGS. 279 

the tie-plate has rocked back and forth until it has worn a 
hole in the tie. Rain-water filling this hole has leached out 
the zinc chloride, and the tie has decayed at this point and 
become unserviceable, when the remainder of the tie showed 
no decay. The creeping of the rails over the ties is sometimes 
the cause of failure of ties which have been effectually secured 
against decay by the use of preservatives. This particular form 
of tie deterioration has been guarded against on a French rail- 
road by using a tie-plate made of creosoted wood, which is 8 
inches long, the same width as the width of the base of the rail, 
and I inch thick. Such wooden plates, which will last a year 
and a half to two years, are made of poplar, or any other hard 
wood, and cost about $2.00 per thousand. It should be ob- 
served that they are used in connection with wooden screws 
instead of the ordinary track spikes. When they are worn 
out, it is only necessary to turn the screw one or two upward 
turns ; the new plate may then be put in endwise and the screw- 
spikes again fastened down. 

A fault 01 the earlier designs of metal tie-plates was that 
they were made of plates which were so thin that they would 
buckle under the pressure of the raiL The claim made for the 
corrugated plates is that their transverse stress is far greater 
than that of a flat tie for the same amount of material; but 
this is not vital provided the flat plates are made sufficiently 
thick so that they will not buckle. The tie-plate used on the 
Southern Pacific Railroad has a slightly beveled surface, the 
plate being | inch thick under the outer edge of the rail and 
^^ inch thick at the inner edge of the rail. 

The holes in a tie-plate should be about -^^ inch larger than 
the size of the intended spike. For example, the holes are 
generally punched with f'Xf" holes for a -j% inch spike, or 
with holes | inch square for a J inch spike. The length of the 
plate (perpendicular to the rail) should be about 3 inches more 
than the base of the rail; this usually means about 8 inches. 
For very heavy traffic, the thickness should be y^ inch to | inch ; 
for average traffic it may be as thin as J inch; some plates are 
made only -^-^ inch thick. For flat-bottom plates the thick- 
ness may be as great as half an inch. The tie-plates under the 
joint ties must be somewhat longer than the intermediates, in order 
to allow for the extra length from out to out of the angle- 
plates. 



280 RAILROAD CONSTRUCTION. § 246. 

246. Method of setting. A very important detail in the 

process of setting the tie-plates on the ties is that the plates 
should be rigidly attached to the ties in their intended position 
during the process of setting. If tie-plates with flat bottoms 
are used, the surface of the tie must be adzed, so that it is not 
only plane but level, so that there will be no danger that the 
plate will rock on the tie. When using tie-plates which are 
corrugated on the under surface, it is necessary to force them 
into the tie until the under side of the plate is flush with the 
surface of the tie. This requires a pressure of several thousand 
pounds. Sometimes trackmen have depended on the easy 
process of waiting for passing trains to force the corrugations 
into the tie until the plate is in its intended position. Until 
the plates are finally set the spikes cannot be driven home, 
and this apparently cheap and easy process generally results 
in loose spikes and rails. The best method for new work is 
to drive the plates into the tie before setting the tie in position. 
A tie-plate gauge holds both tie-plates in their proper relative 
position, and both plates may be driven by the use of heavy 
beetles. When it is necessary to place the plate under the rail 
and drive it in, it is somewhat difficult to drive it by striking 
the plate with a swage on each side of the rail alternately. 
When it is struck on one side, the other side flies up unless held 
down by a wedge driven between the plate and the rail on the 
other side of the rail. A straddler, which straddles the rail 
somewhat like an inverted U, is very useful for this purpose, 
since it makes it possible to strike the head of the straddler and 
force down both sides of the plate at once. The Southern 
Pacific Railroad Company has rigged up a small pile-driver on 
a hand-car, which is used in connection with a straddler to drive 
the tie-plates into position. Some western railroads have even 
adopted the process of rigging up a flat car with a machine 
which will press the tie-plates into place in the ties before the 
ties are placed in the track. 

SPIKES. 

247. Requirements. The rails must be held to the ties by a 
fastening which will not only give sufficient resistance, but which 
will retain its capacity for resistance. It must also be cheap 
tind easily applied. The ordinary track-spike fulfills the last 



,? 247. 



RAIL-FASTENINGS. 



281 



requirements, but has comparatively small resisting power, com- 
pared with screws or bolts. Worse than all, the tendency to 






Fig. 122. 



Fig. 123. 



vertical vibration in the rail produces a series of upward pulls on 
the spike that soon loosens it. When motion has once begun 
the capacity for resistance is greatly reduced, and but little more 
vibration is required to pull the spike out so much that redriving 
is necessary. Driving the spike to place again in the same hole 
is of small value except as a very temporary expedient, as its 
holding power is then very small. Redriving the spikes in new 
holes very soon *' spike-kills " the tie. Many plans have been 
devised to increase the holding power of spikes, such as making 
them jagged, twisting the spike, swelling the spike at about the 
center of its length, etc. But it has been easily demonstrated 
that the fibers of the w^ood are generally so crushed and torn by 
driving such spikes that their holding power is less than that of 
the plain spike. 

The ordinary spike (see Fig. 122) is made with a square cross- 
section which is uniform through the middle of its length, the 
lower 1}" tapering down to a chisel edge, the upper part swelling 
out to the head. The Goldie spike (see Fig. 123) aims to im- 
prove this form by reducing to a minimum the destruction of the 
fibers. To this end, the sides are made smooth, the edges are 
clean-cut, and the point, instead of being chisel-shaped, is ground 



282 



RAILROAD CONSTRUCTION. 



§248. 



down to a pyramidal form. Such fiber-cutting as occurs is thus 
accompHshed without much crushing, and the fibers are thus 
pressed away from the spike and shghtly downward. Any 
tendency to draw the spike will therefore cause the fibers to 
press still harder on the spike and thus increase the resistance. 

248. Driving. The holding power of a spike depends largely 
on how it is driven. If the blows 
are eccentric and irregular in direc- 
tion, the hole will be somewhat en" 
larged and the holding power largely 
decreased. The spikes on each 
side of the rail in any one tie should 
not be directly opposite, but should 
be staggered Placing them direct- 
ly opposite will tend to split the tie, 
or at least decrease the holding 
power of the spikes. The direction 
of staggering should be reversed in 
the two pairs of spikes in any one 
tie (see Fig. 124). This will tend to prevent any twisting of the 
tie in the ballast, which would otherwise loosen the rail from the 
tie. 

249. Screws and bolts. The use of these abroad is very ex- 
tensive, but their use in this country has not passed the experi- 
mental stage. The screws are " wood "-screws (see Fig. 125), 



[_ 



Fig. 124. — Spike-driving. 




Fi<5. 125.— -Screw Spike. 



§ 250. RAIL-FASTENINGS. 283 

having large square heads, which are screwed down with a track- 
wrench. Holes, having the same diameter as the base of the 
screw-heads, should be first bored into the tie, at exactly the 
right position and at the proper angle with the vertical. A 
light wooden frame is sometimes used to guide the auger at the 
proper angle. Sometimes the large head of the screw bears 
directly against the base of the rail, as with the ordinary spike. 
Other designs employ a plate, made to fit the rail on one side, 
bearing on the tie on the other side, and through which the screw 
passes. These screws cost much more than the spikes and re- 
quire more work to put in place, but their holding power is much 
greater and the work of track maintenance is very much less. 
Screw-bolts, passing entirely through the tie, having the head 
at the bottom of the tie and the nut on the upper side, are also 
used abroad. These are quite difficult to replace, requiring that 
the ballast be diig out beneath the tie, but on the other hand the 




Fig. 126. 
occasions for replacing such a bolt are comparatively rare, as 
their durability is very great. The use of screws or bolts in- 
creases the hfe of the tie by the avoidance of " spike-killing." It 
is capable of demonstration that the reduced cost of mainte- 
nance and the resulting improvement in track would much more 
than repay the added cost of screws and bolts, but it seems im- 
possible to induce railroad directors to authorize a large and 
immediate additional expenditure to make an annual saving 
w^hose value, although unquestionably considerable, cannot be 
exactly computed. 

250. "Wooden spikes." Among the regulations for track- 
laying given in § 208, mention was made of wooden '* spikes," 



284 



RAILROAD CONSTRUCTION. 



§250. 



or plugs, which are used to fill up the holes when spikes are 
withdrawn. The value of the policy of filling up these holes is 
unquestionable, since the expense is insignificant compared with 
the loss due to the quick and certain decay of the tie if these 
holes are allowed to fill with water and remain so. But the 
method of making these plugs is variable. On some roads they 
are *' hand-made'' by the trackmen out of otherwise use- 
less scraps of lumber, the work being done at odd mo- 
ments. This policy, while apparently cheap, is not 
necessarily so, for the hand-made plugs are irregular 
in size and therefore more or less inefficient. It is 
also quite probable that if the trackmen are required to 
make their own plugs, they would spend time on these 
very cheap articles which could be more profitably em- 
ployed otherwise. Since the holes made by the spikes 
are larger at the top than they are near the bottom, the 
plugs should not be of uniform cross-section but should 
be slightly wedge-shaped. The ''Goldie tie-plug'' 
(see Fig. 127) has been designed to fill these require- 
ments. Being machine-made, they are uniform in 
size; they are of a shape which w^ll best fit the hole; 
they can be furnished of any desired wood, and at a 
cost which makes it a wasteful economy to attempt 
to cut them by hand. 



Fig. 127. 



TRACK-BOLTS AND NUT-LOCKS. 

251. Essential requirements. The track-bolts must have 
sufficient strength and must be screwed up tight enough to hold 
the angle-plates against the rail with sufficient force to develop 
the full transverse strength of the angle-bars. On the other 
hand the bolts should not be screwed so tight that slipping may 
not take place when the rail expands or contracts with tempera- 
ture. It would be impossible to screw the bolts tight enough to 
prevent slipping during the contraction due to a considerable fall 
of temperature on a straight track, but when the track is curved, 
or when expansion takes place, it is conceivable that the resist- 
ance of the ties in the ballast to lateral motion may be less than 
the resistance at the joint. A test to determine this resistance 
was made by Mr. A. Torrey, chief engineer of the Mich. Cent. 
R. R., using 80-lb. rails and ordinary angle-bars, the bolts being 
screwed up as usual. If required a force of about 31000 to 



§ 252. 



RAIL-FASTENINGS. 



285 



35000 lbs. to start the joint, which would be equivalent to the 
stress induced by a change of temperature of about 22°. But 
if the central angle of any given curve is small, a comparatively 
small lateral component will be sufficient to resist a compression 
of even 35000 lbs. in the rails. Therefore there will ordinarily 
be no trouble about having the joints screwed too tight. The 
vibration caused by the passage of a train reduces the resistance 
to slipping. This vibration also facilitates an objectionable 
feature, viz., loosening of the nuts of the track-bolts. The bolt 
is readily prevented from turning by giving it a form which is 
not circular immediately under the head and making corre- 
sponding holes in the angle-plate. Square holes would answer 
the purpose, except that the square corners in the holes in the 
angle-plates would increase the danger of fracture of the plates. 
Therefore the holes (and also the bolts, under the head) are 
made of an oval form, or perhaps a square form with rounded 
corners, avoiding angles in the outline. 

The nut-locks should be simple and cheap, should have a life 
at least as long as the bolt, should be effective, and should not 
lose their effectiveness w4th age. Many of the designs that have 
been tried have been failures in one or more of these particulars 
as will be described in detail below. 

252. Design of track-bolts. In Fig. 128 is shown a common 
design of track-bolt. In its general form this represents the 

bolt used on nearly all roads, 
being used not only with the 
common angle-plates, but also 
with many of the improved de- 
signs of rail-joints. The varia- 
tions are chiefly a general in- 
crease in size to correspond with 
the increased weight of rails, 
besides variations in detail di- 
mensions which are frequently 
imimportant. The diameter is 
usually f' to |"; V bolts are 
sometimes used for the heaviest 
sections of rails. As to length, 

the bolt should not extend more 
Fig. 128.-Track.bolt. ^^^^ ,„ ^^^^.^^ ^^ ^^^ ^^^ ^^^^^^ 

it is screwed up. If it extends farther than this it is liable to.be 




286 RAILROAD CONSTRUCTION. § 253. 

broken off by a possible derailment at that point. The lengths 
used vary from 3|'', which may be used with 60-lb. rails, to 5", 
which is required with 100-lb. rails. The length required de- 
pends somewhat on the type of nut-lock used. 

253. Design of nut-locks. The designs for nut-locks may be 
divided into three classes: (a) those depending entirely on an 
elastic washer which absorbs the vibration which might other- 
wise induce turning; (6) those which jam the threads of the 
bolt and nut so that, when screwed up, the frictional resistance 
is too great to be overcome by vibration; (c) the ^* positive'' 
nut-locks — those which mechanically hold the nut from turning. 
Some of the designs combine these principles to some extent. 
The ^'vulcanized fiber" nut-lock is an example of the first class. 
It consists essentially of a rubber washer which is protected by 
an iron ring. When first placed this lock is effective, but the 
rubber soon hardens and loses its elasticity and it is then ineffec- 
tive and worthless. Another illustration of class (a) is the use 
of wooden blocks, generally T' to 2" oak, which extend the 
entire length of the angle-bar, a single piece forming the washer 
for the four or six bolts of a joint. This form is cheap, but the 
wood soon shrinks, loses its elasticity, or decays so that it soon 
becomes worthless, and it requires. constant adjustment to keep 
it in even tolerable condition. The "Verona" nut-lock is 
another illustration of class (a) which also combines some of the 
positive elements of class (c). It is made of tempered steel and, 
as shown in Fig. 129, is warped and has sharp edges or points. 
The warped form furnishes the element of elastic pressure when 
the nut is screwed up. The steel being harder than the iron of 
the angle-bar or of the nut, it bites into them, owing to the 
great pressure that must exist when the washer is squeezed 
nearly flat, and thus prevents any backward movement, although 
forward movement (or tightening the bolt) is not interfered 
with. The " National" nut-lock is a type of the second class (6), 
in which, like the " Harvey" nut-lock, the nut and lock are com- 
bined in one piece. With six-bolt angle-bars and 30-foot rails, 
this means a saving of 2112 pieces on each mile of single track. 
The "National" nuts are open on one side. The hole is drilled 
and the thread is cut slightly smaller than the bolt, so that when 
the nut is screwed up it is forced slightly open and therefore 
presses on the threads of the bolt with such force that vibration 
cannot jar it loose. Unlike the " National" nut, the '* Harvey " 



§ 253. 



RAIL-FASTENINGS. 



287 





VERONA 



VULCANIZED FIBRE 




IMPROVED VERONA ^ 




NATIONAL 




Columbia Nut Lock 



C^ZT^ 




JONES 



Fig. 123. — ^Types op Nut-locks. 



288 RAILROAD CONSTRUCTION. § 253. 

nut is solid, but the form of the thread is progressively varied so 
that the thread pinches the thread of the bolt and the frictional 
resistance to turning is too great to be affected by vibration. 

The ''Jones" nut-lock, belonging to class (c), is a type of a 
nut-lock that does not depend on elasticity or jamming of screw- 
threads. It is made of a thin flexible plate, the square part of 
which is so large that it will not turn after being placed on the 
bolt. After the nut is screwed up, the thin plate is bent over so 
that the re-entrant angle of the plate engages the corner of the 
nut and thus mechanically prevents any turning. The metal 
is supposed to be sufficiently tough to endure without fracture 
as many bendings of the plate as will ever be desired. Nut- 
locks of class (c) are not in common use. 



CHAPTER XI. 

SWITCHES AND CROSSINGS. 

SWITCH CONSTRUCTION. 

254. Essential elements of a switch. Flanges of some sort are 
a necessity to prevent ear-wheels from ninning off from the rails 
on which they may be moving. But the flanges, although a 
necessity, are also a source of complication in that they require 
some special mechanism which will, when desired, guide the 
wheels out from the controlling influence of the main-line rails. 
This must either be done by raising the wheels high enough 
so that the flanges may pass over the rails, or by breaking the 
continuity of the rails in such a way that channels or "flange 
spaces'' are formed through the rails. An ordinary stub-switch 
breaks the continuity of the main-line rails in three places, two 
of them at the switch-block and one at the frog. The Wharton 
switch avoids two of these breaks by so placing inclined planes 
that the wheels, rolling on their flanges, will surmount these 
inclines until they are a little higher than the rails. Then the 
wheels on the side toward which the switch runs are guided 
over and across the main rail on that side This rise being ac- 
complished in a short distance, it becomes impracticable to 
operate these switches except at slow speeds, as any sudden 
change in the path of the center of gravity of a car causes very 
destructive jars both to the switch and to the rolling stock. The 
other general method makes a break in one main rail (or both) 
at the switch-block. In both methods the wheels are led to one 
side by means of the ''lead rails," and finally one line of wheels 
passes through the main rail on that side by means of a ''frog." 
There are some designs by which even this break in the main 
rail is avoided, the wheels being led over the main rail by means 
of a short movable rail which is on occasion placed across the 
main rail, but such designs have not come into general use. 

255. Frogs. Frogs are provided with two channel- ways or 
'^ flange spaces" through which the flanges of the wheels move. 

289 



290 RAILROAD CONSTRUCTION. § 255. 

Each channel cuts out a parallelogram from the tread area. 
Since the wheel-tread is always wider than the rail, the wing 
rails will support the wheel not only across the space cut out by 
the channel, but also until the tread has passed the point of the 
frog and can obtain a broad area of contact on the tongue of the 
frog. This is the theoretical idea, but it is very imperfectly 



Fig. 130. — Diagrammatic Design of Frog. 

realized. The wing rails are sometimes subjected to excessive 
wear owing to '^hollow treads'' on the wheels — owing also to 
the frog being so flexible that the point *' ducks" when the wheel 
approaches it. On the other hand the sharp point of the frog 
will sometimes cause destructive wear on the tread of the wheel. 
Therefore the tongue of the frog is not carried out to the sharp 
theoretical point, but is purposely somewhat blunted. But 
the break which these channels make in the continuity of the 
tread area becomes extremely objectionable at high speeds, 
being mutually destructive to the rolling stock and to the frog. 
The jarring has been materially reduced by the device of '' spring 
frogs" — to be described later. Frogs were originally made of 
cast iron — then of cast iron with wearing parts of cast steel, 
which were fitted into suitable notches in the cast iron. This 
form proved extremely heavy and devoid of that elasticity of 
track which is necessary for the safety of rolling stock and 
track at high speeds. The present universal practice is to build 
the frog up of pieces of rails which are cut or bent as required. 
These pieces of rails (at least four) are sometimes assembled by 
riveting them to a flat plate, but this method is now but Httle 
used, except for very light work. The usual practice is now 
chiefly divided between ''bolted" and ''keyed" frogs. In each 
case the space between the rails, except a sufficient flange-way, 
is filled with a cast-iron filler and the whole assemblage of parts 



§ 255. 



SWITCHES AND CROSSINGS. 



291 




Plate VIII. — Some Types of Frogs, 



292 RAILROAD CONSTRUCTION. § 255. 

is suitably bolted or clamped together, as is illustrated in Plate 
YIII. The operation of a spring-rail frog is evident from the 
figure. Since a siding is usually operated at slow speed, while 
the main track may be operated at fast speed, a spring-rail frog 
will be so set that the tread is continuous for the main track and 
broken for the siding. This also means that the spring-rail will 
only be moved by trains moving at a (presumably) slow speed 
on to the siding. For the fast trains on the main line such a 
frog is substantially a ^' fixed" frog and has a tread which is 
practically continuous. 

256. To find the frog number. The frog number (n) equals 
the ratio of the distance of any part on the tongue of the frog 
from the theoretical point of the frog divided by the w^dth of 
the tongue at that point, i.e. =hc^ah (Fig. 130). This value 
may be directly measured by applying any convenient unit of 
measure (even a knife, a short pencil, etc.) to some point of the 
tongue where the width just equals the unit of measure, and then 
noting how many times the unit of measure is contained in the 
distance from that place to the theoretical point. But since c, 
the theoretical point, is not so readily determinable with exacti- 
tude, it being the imaginary intersection of the gauge lines, it 
may be more accurate to measure de, ah, and hs; then n, the frog 
number, =/is -f- (ab + de) . If the frog angle be called F, then 

n=hc-7-ab=hs^(ab-\-de)=i cot §F; 
i.e., cot iF=2n. 

257. Stub switches. The use of these, although once nearly 
universal, has been practically abandoned as turnouts from 
main track except for the poorest and cheapest roads. In some 
States their use on main track is prohibited by law. They have 
the sole merit of cheapness with adaptability to the circum- 
stances of very light traffic operated at slow speed when a con- 
siderable element of danger may be tolerated for the sake of 
economy. The rails from A to B (see Fig. 131 *) are not fastened 

♦The student should at once appreciate that in Fig. 131, as well as in 
nearly all the remaining figures in this chapter, it becomes necessary to use 
excessively large frog angles, short radii, and a very wide gauge in order to 
illustrate the desired principles with figures which are sufficiently small for 
the page. In fact, the proportions used in the figures are such that serious 
mechanical difficulties would be encountered if they were used. These dif- 
ficulties are here ignored because thej^ can be neglected in the proportions 
used in practice. 



! 



§ 257. 



SWITCHES AND CROSSINGS. 



293 



to the ties; they are fastened to each other by tie-rods which 
keep them at the proper gauge; at and back of B they are 




Fig. 131. — Stub Switch. 

securely spiked to the ties, and at A they are kept in place by 
the connecting bar (C) fastened to the switch-stand. One great 
objection to the switch is that, in its usual form, when operated 
as a trailing switch, a derailment is inevitable if the switch is 
misplaced. The very least damage resulting from such a derail- 
ment must include the bending or breaking of the tie-rods of the 
switch-rail. Several devices have been invented to obviate this 
objection, some of which succeed very well mechanicall}', al- 
though their added cost precludes any economy in the total cost 
of the switch. Another objection to the switch is the looseness 
of construction which makes the switches objectionable at high 
speeds- The gap of the rails at the head-block is always con- 
siderable, and is sometimes as much as two inches. A driving- 




FiG. 132. — Point Switch. 



wheel with a load of 12000 to 20000 pounds, jumping this gap 
with any considerable velocity, will do immense damage to the 



294 



RAILROAD CONSTRUCTION. 



§ 258. 



farther rail end, besides producing such a stress in the construc- 
tion that a breakage is rendered quite Hkely, and such a breakage 
might have very serious consequences. 

258. Point switches. The essential principle of a point switch 
is illustrated in Fig. 132. As is shown, one main rail and also 
one of the switch-rails is unbroken and immovable. The other 
main rail (from A to F) and the corresponding portion of the 
other lead rail are. substantially the same as in a stub switch. 
A portion of the main rail (AB) and an equal length of the oppo- 
site lead rail (usually 15 to 24 feet long) are fastened together 
by tie-rods. The end at A is jointed as usual and the other end 
is pointed, both sides being trimmed down so that the feather 
edge at B includes the web of the rail. In order to retain in it 
as much strength as possible, the point -rail 
is raised so that it rests on the base of the ^ 
stock-rail, one side of the base of the 
point-rail being entirely cut away. As 
may be seen in Fig. 133, although the in- 
fluence of the point of the rail in moving 
the wheel-flange away from the stock-rail 
is really zero at that point, 3^et the rail has 
all the strength of the web and about one- 
half that of the base — a very fair angle- 
iron. The planing runs back in straight 
lines, until at about six or seven feet back 
from the point the full width of the head is 
obtained. The full width of the base will only be obtained at 
about 13 feet from the point. An 80-1 b. rail is 5 inches AAide at 
the base. Allowing |" more for a spike between the rails, this 
gives 5}" as the minimum w^dth between rail centers at the 
joint. The minimum angle of the switch-point (using a 15-foot 

5 75 
point-rail) is therefore the angle whose tangent is ' = 

10 X I^ 

.03914, which is the tangent of 1° 50'. Switch-rails are some- 
times used with a length of 24 feet, which reduces the angle of 
the switch-point to 1° 09'. 




Ftg. 133. 



259. Switch-stands. The simplest and cheapest form is the 
"ground lever, ^* which has no target. The radius of the circle 
described by the connecting-rod pin is precisely one-half the 
throw. From the nature of the motion the device is practically 



§ 260. 



SWITCHES AND CROSSINGS. 



295 



self-locking in either position, padlocks being only used to pre- 
vent malicious tampering. The numerous designs of upright 
stands are always combined with targets, one design of which ia 





Fia. 134. — Ground Lever for Throwinq 
A Switch. 



Fia. 135. 



illustrated in Fig. 1»35. When the road is equipped with inter- 
locking signals, the switch-throw mechanism forms a part of the 
design 

260. Tie-rods. These are fastened to the webs of the rails by 
means of lugs which are bolted on, there being usually a hinge- 
joint between the rod and the lug. Four such tie-rods are 



296 



RAILROAD CONSTRUCTION. 



§ 260. 



generally necessary. The first rod is sometimes made with- 
out hinges, which gives additional stiffness to the comparatively 
weak rail-points. The old-fashioned tie-rod, having jaws 
fitting the base of the rail, was almost universally used in the 
days of stub switches. One great inconvenience in their use 
hes in the fact that they must be slipped on, one by one, over 
the jree ends of the switch-rails. Sometimes the lugs are 
fastened to the rail-webs by rivets instead of bolts. 



3. SL ,T7 Tf 




O 



s 



261. Guard-rails. As shown in Figs. 131 and 132, guard-rails 
are used on both the main and switch tracks opposite the frog- 
point. Their function is not only to prevent the possibility of 



§ 262. 



SWITCHES AND CROSSINGS. 



297 



the wheel-flanges passing on the wrong side of the frog-point, 
bnt also to save the side of the frog-tongue from excessive wear. 
The necessity for their use may be realized by noting the apparent 
wear usually found on the side of the head of the guard-rail. 
The flange-way space between the heads of the guard-rail and 
wheel-rail therefore becomes a definite quantity and should equal 
about two inches. Since this is less than the space between 
the heads of ordinary (say 80-pound) rails when placed base to 
base, to sa}' nothing of the f " necessary for spikes, it becomes 
necessary to cut the flange of the guard-rail. The length of the 
rail is made from 10 to 15 feet, the ends being bent as shown 
in Fig. 132, so as to prevent the possibility of the end of the 
rail being struck by a wheel-flange. 



MATHEMATICAL DESIGN OF SWITCHES. 

In all of the following demonstrations regarding switches, 
turnouts, and crossovers, the lines are assumed to represent the 
gauge-lines — i.e., the lines of the inside of the head of the rails. 
262. Design with circular lead-rails. The simplest method 

is to consider that the lead-rails 
curv^e out from the main track- 
rails by arcs of circles which are 
tangent to the main rails and 
which extend to the frog-point F. 
The simple curve from D to F is 
of such radius that (r + ig) vers F 
=^^, in which F=the frog angle, 
C7=gauge, L=the ''lead" (BF), 
and r = the radius of the center of 
the switch-rails. 

9 




j&r.^.. 



Fig. 137. 



r^\g-- 



"vers F' 



(74) 



Also, 
Also, 



BF^BD=cotiF', BD^g; BF = L. 

,', L=g cot iF (75) 

L^(r-h^g)smF; (76) 

QT=2rsmiF (77) 



These formulae involve the angle F, As shown in Table III, 
the angles (F) are always odd quantities, and their trigonometric 
functions are somewhat troublesome to obtain closely with 



298 RAILROAD CONSTRUCTION. § 262. 

ordinary tables. The formulre may be simplified by substitut- 
ing the frog-number n, from the relation that n = JcotJF. 
Since 

r — ^g=L cot F and r + Jg'=L cosec F, 

then r = JL (cot F+ cosec F) 

= ig f*ot J-F(cot F + cosec F) 

= ig cot^ JF, since (cot a + cosec a) =cot Ja 

i=2gn^ (78) 

Also, L=2gn, (79) 

from which r = nXL (80) 

These extremely simple relations may obviate altogether the 
necessity for tables, since they involve only the frog-n\nnber and 
the gauge. On account of the great simplicity of these rules, 
they are frequently used as they are, regardless of the fact that 
the curve is never a uniform simple curve from switch- block to 
frog. In the first place there is a considerable length of the 
gauge-Hne within the frog, which is straight unless it is pur- 
posely curved to the proper curve while being manufactured, 
which is seldom if ever done — except for the very large-angled 
frogs used for street-railway work, etc. It is also doubtful whether 
the switch-rails (BA, Fig. 181) are bent to the computed curve 
when the rails are set for the switch. The s\^dtch-rails of point 
switches are straight, thus introducing a stretch of straight track 
which is about one-fifth of the total length of the lead-rails. The 
effect of these modifications on the length and radius of the lead- 
rails will be developed and discussed in the next four sections. 

The throw (t) of a stub switch depends on the weight of the 
rail, or rather on the width of its base. The throw must be at 
least J" more than that wndth. The head-block should there- 
fore be placed at such a distance from the heel of the sT\dtch (B) 
that the versed sine of the arc equals the throw. These points 
must be opposite on the two rails, but the points on the two rails 
where these relations are exactly true will not be opposite. 
Therefore, instead of considering either of the two radii (r-\-hg) 
and (r — iy), the mean radius r is used. Then (see Fig. 137) 

vers KOQ=t-^r, 

and the length of the switch-rails is 

QK=r sin KOQ (81) 



§ 264. 



SWITCHES AND CROSSINGS. 



299 



Stub-switches are generally used with large frog angles. For 
small frog angles (large frog-numbers) the values of QK are so 
great that the length of rail left unspiked is too great for a safe 
track. If this were obviated by spiking down a portion of the 
lead the theoretical accuracy of the switch would be lost. The 
values of QK for various frog-numbers is given in Table III. 
These are based on a uniform throw {t) of 5^ inches. 

263. Effect of straight frog-rails. A portion of the ends of 

the rails of a frog are free and may 
be bent to conform to the switch- 
rail curve, but there is a consid- 
erable portion which is fitted to 
the cast-iron filler, and this por- 
tion is always straight. Call the 
length of this straight portion 
back from the frog-point / ( =FH, 
Fig. 138). Then we have 
r-^ig = (g —f sin F) ~ vers F 

=— ^-/cot^F 
vers F ' 



\ 







FH=/ 




FiQ. 138. 



g 



vers F 



-2/n. 



(82) 



BF=L = (g-f sin F) cot iF+f cos F 
= 2gn-f sin F cot iF + / cos F 
= 2gn-f(l+eos F) +/ cos F 
= 2gn-f 



(83) 



Since r — ig = (L—f sec F) cot F, and 
r-\-\g = {L—f cos F) cosec F, 
r = iL(cot i^ + cosec F) —if sec F cot F — hf cos F cosec F 



^' \ sui / / 



r=Ln-J/cotiF 

=Ln—fn. Then from (83) 
r=^2gn}~2fn 



(84) 



264. Effect of straight point-rails. The "point switches,*' 
now so generally used, have straight switch-rails. This requires 



300 



RAILROAD CONSTRUCTION. 



264. 



an angle in the alignment rather than turning off by a tangential 
curve. The angle is, however, very small (between 1° and 2°), 
and the disadvantages of this angle are small compared with 
the very great advantages of the device. 




"2sini(^ + a) sin J(-^-a) 
g—k 



cos a — cos F' 



(85) 



BF = L= FM cos \{F f a) + DN 
=(g-k) coti(F + a)+DN. . 



(86) 



265. Combined effect of straight frog-rails and straight point- 
rails. It becomes necessary in this case to find a curve which 
shall be tangent to both the point-rail and the frog-rail. The 
curve therefore begins at M, its tangent making an angle of a 
(usually 1° 50') with the main rail, and runs to H. The central 



§ 266. 



SWITCHES AND CROSSINGS. 



301 



angle of the curve is therefore (F — a). The angle of the chord 
HM with the main rails is therefore 



UF-a)+a = KF + a); 



HM = 



r-\-ig = 



g—f sin F—k ^ 
sinK^ + a) ' 

HM 
2sini(F-a) 

^—/ sin F—A; 

" 2 sin i(F + a) sin i(F—a) 
g—f sin F — k ^ 
' cos a— cos F' 

ST = 2r sin i(F -a). . . 



(87) 
(88) 



BF=L=HM cos h{F + a) +/ cos F+DN 

= (g-f sin F-k) cot i(F + a) +/ cos F-hDN. . . (89) 

It may be more simple, if (r-hig) has already been computed, 
to write 

L =2(r + ig) sin i{F-a) cos i(F + a) +/ cos F+DN 
^{r + ig){sm F-sin a) -{-f cos F+DN (90) 



V-R 




FiQ. 140. 



266. Comparison of the above methods. Computing values 
for r and L by the various methods, on the uniform basis of a 



302 KAILROAD CONSTRUCTION. § 266. 

No. 9 frog, standard gauge 4' 8 J'', / = 3'.37, /:=5|-" =0'.479, 
DN = 15' 0'^, and a = l° 50', we may tabulate the comparative 
results: 



§ 262. 
Simple circle. 
Curved frog- 
rail. Curved 
switch-rail. 



§263. 

Straight 

frog-rail. 

Curved 

switch-rail. 



§264. 
Curved frog- 
rail. Straight 
switch-rail. 



§265. 

Straight 

frog-rail. 

Straight 

switch-rail. 



Deg. of curve 
L 



762.75 
7° 31' 
84.75 



702.00 
S° 10' 
81.37 



747.48 
7° 40' 
74.00 



681.16 
8° 25' 
72.13 



This shows that the effect of using straight frog-rails and 
straight switch-rails is to sharpen the curve and shorten the lead 
in each case separately, and that the combined effect is still 
greater. The effect of the straight switch-rails is especially 
marked in reducing the length of lead, and therefore Eq. 78 to 
80, although having the advantage of extreme simplicity, can- 
not be used for point-switches without material error. The 
effect of the straight frog-rail is less, and since it can be mate- 
rially reduced by bending the free end of the frog-rails, the in- 
fluence of this feature is frequently ignored, the frog-rails are 
assumed to be curved, and Eq. S5 and 86 are used. (See § 276 
for a further discussion of this point.) 

267. Dimensions for a turnout from the OUTER side of a curved 




track. In this demonstration the switch-rails will be considered 
as uniformly circular from the switch-points to the frog-point. 



§ 267. SWITCHES AND CROSSINGS. 303 

In the triangle FCD (Fig. 141) we have 
(FC-hCD) \{FC-CD)\'.im ^(FDC + DFC) :tan i(FDC-DFC); 
but HFDC + DFC) = 90° - ^6 

and i(FDC-DFC)=iF. 

Also, FC + CD=2R and FC-CD=^g; 

.-. 27? :^ -.-cot §<9 ttanJF 
:: cot JF : tan J/9; 

.-. tanj§ = ^ (91) 

Also, OF : FC :: sin <9 : sin ^; but (f>=^{F-d)] 

then , + j^ = (7^ + i^)_^|l_ (92) 

5F=L = 2(/? + ig)sin J^ (93) 

If the curvature of the main track is very sharp or the frog 
angle unusually small, F may be less than d ; in which case the 
center will be on the same side of the main track as C. Eq. 
92 will become (by calling r= —r and changing the signs) 

(f-hg)=iR+ho^^^ (94) 

If we call d the degree of curve corresponding to the radius 
r, and D the degree of curve corresponding to the radius B,, also 
df the degree of curve of a turnout from a straight track (the frog 
angle F being the same), it may be shown that d=d' —D (very 
nearly). To illustrate we will take three cases, a number 6 
frog (very blunt), a number 9 frog (very commonly used), and a 
number 12 frog (unusually sharp). Suppose D=4° 0'; also 
D = 10° 0'; g=^' 8V'=4'.708. 

A brief study of the tabular form on p. 279 will show that the 
error involved in the use of the approximate rule for ordinary 
curves (4° or less) and for the usual frogs (about No. 9) is really 
insignificant, and that, even for sharper curves (10° or more), 
or for very blunt frogs, the error would never cause damage, 
considering the lower probable speed. In the most unfavorable 
case noted above the change in radius is about 1%. On account 
of the closeness of the approximation the method is frequently 
used. The remarkable agreement of the computed values of L 



304 



RAILROAD CONSTRUCTION. 



267 



Frog 




Z) = 4°. 




"L" for 
straight 
track. 


num- 
ber. 


d 


d'-D 


Error. 


L 


6 

9 

12 


12° 54' 20" 
3 30 27 
13 33 


12° 57' 52" 
3 31 04 
13 36 


0° 03' 32" 
37 
03 


56.57 

84.85 

112 72 


56.50 

84.75 

113.00 



Frog 


D = 10°. 


♦•L" for 
straight 
track. 


num- 
ber. 


d 


d'-D 


Error. 


L 


6 

9 

12 


6° 53' 24" 
2 27 54 
5 44 26 


6° 57' 52" 
2 28 56 
5 46 24 


0° 04' 28" 
01 02 
01 58 


56.66 

84.86 

112.91 


56.50 

84.75 

113.00 



with the corresponding values for a straight main track (the lead 
rails circular throughout) shows that the error is insignificant in 
using the more easily computed values. 

268. Dimensions for a turnout from the INNER side of a curved 
track. (Lead rails circular throughout.) From Fig. 142 we 
have, from the triangle DFC, 




DF-\-FC:DF-FC ::tsLni(DFC + FDC):tsLni(DFC-FDO; 
but h(DFC + FDC) = 90° - J^ 

and i{DFC-FDC)=iF; 

.'. 2R : ^ ::cot \d : tan JF 
:* cot \F tan hd] 



tani^ = ^. 



(95) 



§ 269. 



SWITCHES AND CROSSINGS. 



305 



From OFC, 



OF:FC:\smQ:sm(F+d), 



^ sin (F + ey 
L=BF=2(R-ig)smid, 



(96) 
(97) 



As in § 267, it may be readily shown that the degree of the 
turnout (d) is nearly the sum of the degree of the main track (D) 
and the degree (d^) of a turnout from a straight track when the 
frog angle is the same. The discrepancy in this case is some- 
what greater than in the other, especially when the curvature 
of the main track is sharp. If the frog angle is also large, the 
curvature of the turnout is excessively sharp. If the frog angle 
is very small, the liability to derailment is great. Turnouts to 
the inside of a curved track should therefore be avoided, unless 
the curvature of the main track is small. 

269. Double turnout from a straight track. In Fig. 143 the 
frogs Fi and Fr are generally made equal. Then, if there are 




uniform curves from B^ to Fi and from ^ to Fr, the required 
value of Fjn is obtained from 



vers iFm = 



2(r + W 



(98) 



r being found from Eq. 78, in which n is the frog number of Fi 
or Fr. 

MFm=r tan JFm; 

but since rim = J cot iFm, 



MFm-= 



2nn 



(99) 



306 RAILROAD CONSTRUCTION. § 289. 

Since vers Fi=^-. — r^i"^; 

vers iFm = iyeTsFi (100) 

Also, since (CiFmy = {MFmy + (C,M)\ we have 

Simplifying and substituting, r = 2gn'^, we have 

4ghi\ 
'Ann 



2ghi^ + ig^=l^ 3, 



"""^ ~2n2 + r 
Dropping the \, which is always insignificant in comparison with 
2n^, we have 

nm=-77s=nX.707(approx.) (101) 

Frogs are usually made with angles corresponding to integral 
values of n, or sometimes in ''half" sizes, e.g. 6, 6 J, 7, 7 J, etc. 
If No. 8J frogs are used for Fi and Fr, the exact frog number 
for Fm is 6.01. This is so nearly 6 that a No. 6 frog may be used 
without sensible inaccuracy. Numbers 7 and 10 are a less 
perfect combination. If sharp frogs must be used, 8 J and 12 
form a very good combination 

If it becomes necessary to use other frogs because the right 
combination is unobtainable, it may be done by compounding 
the curve at the middle frog. Fi and Fr should be greater 
than JFm. If equal to \Fm, the rails would be straight from 
the middle frog to the outer frogs. In Fig. 144, d^=Fi — \Fm' 



Drawing the chord FiFm, 

KFiFm = Fi-id,=Fi-iFi-^iFm = i(Fi + hFm); 

KTra _ g 



^^^" '^^^WW^n. " 2 Sin ^{Fl + Wm) ' • ' ^'^'^ 

KFi=KFr^^QoiKFiFm = hgGoti{Ft + hFm)) . (103) 



2 sin iO 4 sin ^{Fi + iFm) sin i{Fi -}Fm) 

^ .... (104) 



cos ^Fm— COS Fl 



§ 270. 



SWITCHES AND CROSSINGS. 



307 



If three frogs, all different, must be used, the largest ma}^ be 
selected as Fm ; the radius of the lead rails may be found by an 
inversion of Eq. 98; Fm may be located in the center of the 
tracks by Eq. 99; then each of the smaller frogs may be located 




Fig. 144. 

by separate applications of Eq. 102 or 103, the radius being 
determined by Eq, 104. 

270. Two turnouts on the same side. In Fig. 145, let Oj 
bisect O2D, Then (rj + J^) = J(^2 + iS') ; ^^so, Ofli=O^Fi smd 
Fr^Fi, 

5^ _ 2sr . 



vers Fv 



(105) 



^i^m =(r' + j9')sini^m (106) 

It may readily be shown that the relative values of Fr, Fi, and 
Fm are almost identical with those given in § 269; as may be 





\N 


fr 




<^y^ 


\ 




^Xpy VN 


.\ 


^ 


.^^ / ^ \ \ 

// w 


\ 


^ 


A' 1 


] 


u 


Oi 


B 



Fig. 145. 



apparent when it is considered that the middle switch may be 
regarded simply as a curved main track, and that, as develpped 



308 



RAILROAD CONSTRUCTION. 



§271. 



in § 267, the dimensions of turnouts are nearly the same whether 
the main track is straight or slightly curved. 

271. Connecting curve from a straight track. The "con- 
necting curve" is the track 
lying between the frog and 
the side track where it be- 
comes parallel to the main 
traick. (FS in Fig. 146 or 147). 
Call d the distance between 
track centers. The angle 
FO,R=F (see Fig. 146). 
Call r' the radius of the con- 
necting curve. Then 
d-g , 




(r'-ig) = 



vers F ' 



(107) 
FR=(r'- 



Fig. 146. 



■ig) sini^. 



(108) 



If it is considered that the distance FR con^mes too much 
track room it may be shortened by the method indicated in 
Fig. 151. 

272. Connecting curve from a curved track to the OUTSIDE. 
When the main track is curved, the required quantities are the 




Fig. 147. 

radius r of the connecting curve from F to S, Fig. 147, and its 
length or central angle. In the triangle CSF 

CS + CF'.CS-CFr.tsLU i(CFS-^CSF):tsin i(CFS -CSF); 



§ 273. SWITCHES AND CROSSINGS. 309 

but i(CFS-^CSF) =90 -i(p; and, since the triangle O^SF is 
isosceles, i(CFS-CSF)=hF; 

.-. 2R-\-d:d-g::coti(p:tsiniF 
::cot JF:tan i<(f; 

■■■ '"W'-i^ <™' 

From the triangle CO^F we may derive 

r — hg:R-i-ig:: sin (/f • sin (F -h(p); 

^-*^ = (^-^i^)swf?) ^'''^ 

Also FS = 2(r-ig)smi(F-^(/;) (Ill) 

273. Connecting curve from a curved track to the INSIDE, 




Fio. 148. 

As above, it may readily be deduced from the triangle CFS (see 
Fig. 148) that 

(2R-d):{d-g)::cot J^:tan JF, 

and finally that 

2n(d-g) 



tan J^ = 



2R-d 



Similarly we may derive (as in Eq. 110) 



(112) 



(113) 



310 



RAILROAD CONSTRUCTIOX. 



§273. 



Also 



FS = 2(r-ig)smi{F-<P) (114) 



Two other cases are possible 
becomes infinite (see Fig. 149), 
then F = (p. In such a case 
we may write, by substitut- 
ing in Eq. 112, 

2R-d = 4n\d-g), . (115) 

This equation shows the value 
of R, which renders this case 
possible with the given values 
of n, d, and g. (h) ^ may be 
greater than F. As before 
(see Fig. 150) 

2R—d:d—g::cot iipitanhF; 
the same as Eq. 112, but 



T-¥hg = {R-h9) 



(a) r may increase until it 




Fig. 149. 



sin (/f 
sin{<p--F)' 



(116) 




Fig. 150. 



Problem. To find tlie dimensions of a connecting curve run- 
ning to the inside of a curved main track; number 9 frog, 4° 30' 
curve, d==13', g=A' 8 J". 



§ 274. 



SWITCHES AND CROSSINGS. 



311 



Solution, 

Eq. 112. 

(± 

B = 1273.6~" 
2« = 2547.2 
2i2-d = 2534.2 
log(2jB--d) = 3.40384 



d=13.000 

(7 = 4.708 

-g)= 8.292 



Eq. 116. 






1273.6 
2.35 



(R-^g) =1271.25 

(♦-2?) = 1373''. log = 3. 13767 

4.68557 

log sin (*-F) = 7.82324 

Eq. 114. 

^(*-i?') = 686."5 ..2.83664 

4.68557 

sin K*-F) =7.52221 



log 2n = 

log (d-g) = 

co-log (2R-d) = 

log tan i^ = 

(^-F) = 



=1.25527 
= .91866 
= 6.59616 
=8.77009 
3° 22' 14" 
go 44/ 28'/ 

6° 21^ 35 ^^ 
0° 22' 53' 



iog(i2-^a) = 3.10423 

log sin * = 9. 06960 

co-log sin (*-F)-2JT676 

(r + ^^) = 22418.0. .4.35059 

r = 22415.6 

rf = 0° ly 

2... 0.30103 

(r + ifi^) = 22418.0...4.35059 

sini(Sk-i0... 7.5222r 

F>S=149.22. 2. 17384 



274. Crossover between 



§! 






V 








^ 


3^^. 


. 




— 




1\ 


re\ 




Q / 
j 


1 








\ 


\ 




^11 












r 










\ 


/ 




Y 


\ 








1. —d— 


i 


^ 


K 






° 






Oj 



FiQ. 151. 



two parallel straight tracks. (See 
Fig. 151.) The turnouts 
are as usual. The cross- 
over track may be straight, 
as shown by the full lines, 
or it may be a reversed 
curve, as shown by the 
dotted lines. The reversed 
curve shortens the total 
length of track required, 
but is somewhat objection- 
able. The first method re- 
quires that both frogs must 
be equal. The second 
method permits unequal 
frogs, although equal frogs 



are preferable. The length of straight crossover track is F^T. 



FyT sin F^^g cos F^ =d-g; 



^'^=^-^-<^^>- 



(117) 



312 



RAILROAD CONSTRUCTION. 



§274. 



The total distance along the track may be derived as follows: 

DV ^2DF, + F^Y = 2DF, + XY -XF^) 
XY = (d-g) cotFi; XF^=g^s\nF2) 



DV=2DF^ + {d-g) cot F^- 



sin F. 



If a reversed curve with equal frogs is used, we have 



also 



vers = — : 
2r 



DQ=2rsmd 



(118) 

(119) 
(120) 




FiQ. 152. 

If the frogs are unequal, we will have (see Fig. 152) 
r, vers d+r^ vers d=d; 

/. vers^ = — --; . . . . 

^ + ^2 

also the distance along the track 

B2N = (ri-\-r2) sind. . . . 



(121) 



(122) 



Problem, A crossover is to be placed between two parallel 
s\^ght tracks, 12' 2" between centers, using a No. 8 and a No. 9 



§274. 



SWITCHES AND CROSSINGS. 



313 



frog, and with a reversed curve between the frogs. Required 
the total distance between switch-points (the distance ^2^ ^^ 
Fig. 152). 

Solution. If straight point rails and straight frog rails are 
used, the radii, r^ and r2, taken from the middle section of Table 
III, are 527.91 and 681.16. 



Eq. 122. 

n = 527.91 
r2 = 681.16 
r, 4- r2= 1209.07 

Eq. 122. 



vers 6 
d=\2' 2" =12.16, 

(9 = 8*^ 08' 06" 



d_ 

"n + ra 

log =1.08517 
logrr|+r2) = 3.08245 
log vers ^ = 8.00272 



BiN=^\l\.m 



log(ri + r2) =3.08245 
log sin ^ = 9.15077 
log 171.09 = 2.23322 



The length of the curve from 52 = 100(/9-4-<i) =100(8° 08' 06" -r- 
8° 250 =96.65. The length of the other curve is 100(8° 08' 06' -J- 




FiQ. 153. 



10° 52') =74.86. As a check, 96.65 + 74.86 = 171.51, which is 
slightly in excess of 171.09, as it should be. 



314 



RAILROAD CONSTRUCTION. 



§275. 



275. Crossover between two parallel curved tracks, (a) Using 
a straight connecting curve. This solution has limitations. If 
one frog (F^) is chosen, F2 becomes determined, being a function 
of jPj. If Fi is less than some limit, depending on the width (d) 
between the parallel tracks, this solution becomes impossible. 
In Fig. 153 assume F^ as known. Then FJI=g sec F^, In the 
triangle HOF2 we have 

sin HF2O : sin FMO wHO'.Ffi) 
sin F^HO = cos F, ; HFjO = 90° f F. ; 
.*. sin HF20=cos F2. 
HO=R + id-ig-g sec F^] F20=R-\d^\g] 
R-^-^d — ^g—g sec F^ 

R-\d + Yg 



cos F2 = cos F^- 



(123) 



Knowing F2, 62 is determinable from Eq. 91. Fig. 153 shows 
the case where 62 is greater than F2. Fig. 154 shows the case 
v\^here it is less. The demonstration of Eq. 123 is applicable to 




Fig. 154. 



both figures The relative position of the frogs Fj and F2 may 
be determined as follows, the solution being applicable to both 
Figs. 153 and 154: 

/f Oi^2 = 180° - (90° -i^i) - (90° + i^2) =^1 --^2. 



Then 



GF,=2(R + id-^g)smi(F,-F2) (124) 






Since F2 comes out any angle, its value will not be in general 
that of an even frog number, and it will therefore need to be 
made to order. 



§275. 



SWITCHES AND CROSSINGS. 



315 



(b) Continuing the switch-rail curves until they meet as a 
reversed curve. In this case Fi and F2 may be chosen at pleasure 
(within limitations), and they will of course be of regular sizes 
and equal or unequal as desired. F^ and F2 being known, d^ 
and 62 are computed by Eq. 95 and 91. In the triangle 00^2 
(see Fig. 155) 

2(S-OO2)(S-OO0 
vers^= (002)(00i) 

in which S = i(00, + OO2 + Ofii) ; 

but 00,=R + id-r,, 

002=R-id+r2y 

/. S = i(^R-h2r2)=R+r2; 
S-002 = R-\-r2-R-{-id-r2 = id; 
S-OOi=R + r2-R-^d + ri=r^-]-r2-id; 




vers0 = 



Fig. 155. 
djr.-hr.-jd) 



{R-id + r2XR + id-ri) ' 



- nr^r, • ,00i . .R + id-r. 
sinOOjOi^sm (/fyr-~=sin ip ^ J; 



OA^=^+OA^; 

NF2-=2{R-\d-\-\g) sin ^{^j; -6,-62), 



(125) 

(126) 

(127) 
(128) 



316 RAILROAD CONSTRUCTION. § 275. 

Although the above method introduces a reversed curve, yet 
it uses up less track than the first method and permits the use of 
ordinary frogs rather than those having some special angle which 
must be made to order. But the above solution impUes the use 
of circular lead rails. We may compute dimensions and lay 
track between F^ and F^ on this basis and then change the 
switch rails as desired. Strictly, r^ and r^ should be computed 
by Eq. 92 and 96, but for an easy main-line curve the approxi- 
mate rule is sufficiently accurate. 

Problem. — Required the dimensions of a crossover on a 
4° 30' curve when the distance between track centers is 13 feet. 
The frog for the outer main track (F^ in Fig. 155) is No. 9; 
F^ is No. 7. Then 22=1273.6; R^y for the outer main track, 
= 1280.1; Z)i = 4°29'; 7^2=1267.1; D2=4°3r; ri= radius for 
(di + A)"" curve = radius for (7° 31^ + 4° 29') curve = 478.34; 
r2= radius for {d^-D^Y curve = radius for (12° 26' -4° 31? 
curve=724.31. (See §§ 267-268.) 



Eq. 125 
ri+r2-^d = 1196.15 
/iJ-irf + r2 = 1991.31 
R + \d-r^= 801.76 
il> = 7° 52' 26 


f 
tan 


d = lS 

log = 3. 30914 
log = 2.9040i 

log = 3. 08013 
OOoOi^S" 14' 24" 


log( 
°06' 


log = l. 11394 

log = 3. 07778 

colog = 6. 69086 

colog = 7 09595 

log vers ^ = 7. 97854 


Eq. 126 
n+rj- 1202.65 


log sin v^= 9. 13670 

R + *rf-ri)=2.9040i 

colog = 6. 91986 

sinOO2Oi = 8.96061 


Eq. 127 O2P1D 


» 52' 26" + 5° 14' 24" = 13 


^" 










Eq. 91 

^,=3'»47' 30" 


gn 4.708X9 42.372 
- ^ R 1280.1 ' 1280.1 

(Using Table VI) 
^^1 = 1° 53' 45" 

gn 4.708X7 32.956 
R 1267.1 1267.1 

i^2 = l** 29^24" 
R-id-hitg^l269A5 


log = l. 62708 

log = 3. 10724 

logtani^i = 8. 51983 

5-31426 

log 6825 = 3 -83410 


Eq. 95 

* ^2=-2*>58'48" 


log = l. 51795 

log = 3. 10281 

tan i^2 = 8 -41512 

5-31433 

log 5364 = 3 -72945 


Eq. 128 


2 ' log = -30103 
log = 3. 10361 


NF2^4S.S4: 


= 1^06' 08" 




log sin =4 -68555 

^ 3.59857 

V»g 48. 84 = 1. 68876 



? 276. SWITCHES AND CROSSINGS. 317 



Length of curve with radius Vi = 100 ~T9^"7y — = 109. 18 

5° 14' 24" 
•♦ *' •* " r2 = 100 ^o ,rr = 66.16 



Total length of curve between switch points = 175.34 

As an approximate check, the mean length subtending the 
angle ^ with radius R is similarly computed as 174.98. Note 
that the length of the curve with the radius r, is 66.16, which 
is but Uttle more than the length of lead rails (65.92) for a No. 7 
frog using circular lead rails, which means that the point of 
reversed curve is but little beyond the frog point. If the 
computations had apparently indicated the point of reversed 
curve coming be ween the frog point and the switch point, it 
would have showTi the impracticability of the combination 
of No. 7 and No. 9 frogs with this particular degree of curve^ 
gauge of track, and distance between track centers. If both 
frogs were made No. 9, the total length of track between sw* ch 
points would be increased to over 198 feet and the point of 
reversed curve would be nearly at the middle point. This 
shows that the frog numbers should be nearly equal, but also 
shows that there is some choice '^ within limitations.' ' 

276. Practical rules for switch-laying. A consideration of 
the previous sections will show that the formulae are compara- 
tively simple when the lead rails are assumed as circular; that 
they become complicated, even for turnouts from a straight 
main track, when the effect of straight frog and point rails is 
allowed for, and that they become hopelessly complicated when 
allowing for this effect on turnouts from a curved main track. 
It is also shown (§ 267) that the length of the lead is practically 
the same whether the main track is straight or is curved with 
such curves as are commonly used, and that the degree of curvQ 
of the lead rails from a curved main track may be found with 
close approximation by mere addition or subtraction. From 
this it may be assumed that if the length of lead (L) and the 
radius of the lead rails (r) are computed from Eq. 87 and 90 for 
various frog angles, the same leads may be used for curved main 
track; also, that the degree of curve of the lead rails may be 
found by addition or subtraction, as indicated in § 267, and that 
the approximations involved will not be of practical detriment. 
In accordance with this plan Table III has been computed from 



318 



RAILROAD CONSTRUCTION. 



§ 276. 



.^ 



MN = fc 
, FH=/ 
VMDN=a 
\/HMR=^(F-a) 




Fig. 140. 



Eq. 87, 88, and 90. The leads there given may be used for all 
main tracks, straight or curved. The table gives the degree of 
curve of the lead rails for straight 
main track; for a turnout to the 
inside, add the degree of cur^^e of the 
main track; for a turnout to the out- 
side, subtract it. 

If the position of the switch-block 
is definitely determined, then the 
rails must be cut accordingly; but 
when some freedom is allowable 
(which never need exceed 15 feet 
and may require but a few inches), 
one rail-cutting may be avoided. 
Mark on the rails at B, F, and D; 
measure off the length of the switch- 
rails DN and locate the point M at 
the distance k from A^. If the frog 
must be placed during the brief period between the running 
times of trains, it will be easier to joint up to the frog a piece 
of rail at one or both ends of just such a length that they may 
be quickly substituted for an equal length of rail taken out of 
the track. When the frog is thus in place the point H 
becomes located. The chord MH may be measured on the 
ground. The curve between M and H is of known radius. 
Substituting in Eq. 31 the value of chord and R, 
we may compute x ( = db). Locate the middle 
point d and the quarter points a'^ and c'^. Then 
a^^a and c^^c each equal three-fourths of db. Theo- 
retically this gives a parabola rather than a circle, 
but the difference for all practical cases is too 
small for measurement. 

Example. — Given a main track on a 4° curve — 
a turnout to the outside, using a No. 9 frog; 
gauge 4' Sy'; /=3^37; k=5r; DN=15' 0'' and 
o= 1° 50'. Then for a straight track r would equal 
681.16 [d=S° 25^. For this curved track d will 
be nearly (8° 25' -4°) = 4° 25', or r will be 1297.6. 
L for the straight track would be 72.20; but since 
the lead is slightly increased (see § 267) when the turnout is 
on the outside of a curve, L may here be called 72.5. H and 



I 



H6 



/S 



Fig. 156- 



§277. 



SWITCHES AND CROSSINGS. 



319 



M may be located as described above. MH may be measured 
on the ground, or since it will be in this case about 0.10 longer 
than the computed value oi ST (=53.80) given in Table III, 
and since it is slightly more for a turnout to the outside of a 

(54.0)2 



curve, it may be called 54.0. Then x=db 
feet, and aa^^ and cc^^ = 0.21 foot. 



8X1299.95 



= 0.280 



CROSSINGS. 

277. Two straight tracks. When two straight tracks cross 
each other, four frogs are necessary, the angles of two of them 




being supplementary to the angles of the other. Since such 
crossings are sometimes operated at high speeds, they should be 



320 



RAILROAD CONSTRUCTION. 



§278. 



Structurally the 



very strongly constructed., and the angles should preferably be 
90° or as near that as possible. The frogs will not in general 
be ''stock'' frogs of an even number, especially if the angles are 
large, but must be made to order with the required angles as 
measured. In Fig. 157 are shown the details of such a crossing. 
Note the fillers, bolts, and guard-rails. 

278. One straight and one curved track, 
crossing is about the same as above, 
but the frog angles are all unequal. 
In Fig. 158, R is known, and the 
angle M, made by the center lines 
of the tracks at their point of inter- 
section, is also known. 

M = NCM. XC = R cos M. 
(R-^g) cos F,=XC + ig; 



cos F^ = 



R cos M 4- ig 



Similarly 



cos F2 = 

cos Fg = 

cos -P4 = 



R 


R-hj ' 
cos M + J<7 


R 


R+hg 

cos M — 


1 

\g 


R 


R + hg 
cos M — 


> 

■hg 



R-ig 



(129) 




Fig. 158. 



tSini{Ci-C2)=cotiM 
Ci and Cj then become known and 

^ ^ ^sm Cj 



R2-\-Ri 



(131) 



(132) 



F^F, = {R + hg) sin F,-{R-ig) sin F,; ) 

HF, = (R-ig)(smF,-smF,), f ' ' ^'"'^^ 

279. Two curved tracks. The four frogs are unequal, and j 
the angle of each must be computed. The radii R^ and R2 are ' 
known; also the angle M. r^, r^, rj, and r^ are therefore known ^ 
by adding or subtracting hg, but the lines are so indicated for 
brevity. Call the angle MCiC2 = Ci, the angle MCiC^^Ci, and 
the line CiC2=c. Then 

J(Ci + C2)=90°-iM 
and 

fR2—R\ 



§ 279. SWITCHES AND CROSSINGS. 321 

In the triangle Ffifiiy call i(c + ^i + 0=5i; S2 = i(c + raH-rJ; 




^3 = 4(^ + ^1 + ^3); and 54 = ^(^ + ^2 + ^3)- Then, by formula 29, 
Table XXX, 



Similarly 



vcia 


•^'1 




TiTi 






F, 


2{8,- 


-7-2) (S2- 


-rd 






^2^4 






F,- 


2GS3- 


-n)(S3- 


~n) 






^1^3 


i 


TT-Q-ro 


w . 


2(5,- 


-^2)(54- 


-n) 



(133) 



^2^2 

sin Ci 02^4 = sin i^,-; 

sinCiC2^2=sini^2^; 

.-, F^C^F.^Cfi^F^-Cfi^F.,, (134) 

sini^iCA=sini^i^; 

sinXCA = sinF2-, 

.-. Ffi,F,=F,Cfi,-F,Cfi,', (135) 

from which the chords F^F.^ and ^2^4 ^re readily computed. 



322 



RAILROAD CONSTRUCTION. 



§279. 



F1F2 and ^2^4 are nearly equal. When the tracks are straight 
and the gauges equal, the quadrilateral is equilateral. 

Problem. Required the frog angles and dimensions for a cross- 
ing of two curves (i)i=4°; 1)2 = 3°) when the angle of their tan- 
gents at the point of intersection =62° 28' (the angle M in 
Fig. 159). 

Solution 

i?i = 1432.7; /?2 = 1910.1; 

n =J?2 + i^ = 1910. 1+2. 35 = 1912. 45; 

Ti =/^,-is^ = 1910. 1-2. 35 = 1907.75; 

rg =i?i + i9r = 1432. 7 + 2. 35 = 1435. 05; 

n =/?i-Jy = 1432. 7-2. 35 = 1430. 35. 
Eq. 131. log cot JM =0.21723 

R,-R,=A71A] log =2.67888 

i^^ + i?! =3342.8; log =3.52411; co-log = 6 . 47589 



i(cl 


-Cj) =13° 15' 07''; tan 13° 15' 07" =9. 37200 


i(Ci + C2)=58°46' 


[J(Ci + C2)=90°-iM] 




Ci = 72°01'07 


r// 




C, = 45°30'53" 


Eq. 132. 




logi?2=3.28105 
log sin M =9.94779 




log sin C] 


= 9.97825; co-log =0.02175 


c = C,C2 = 1780.7; 


logCiC2=3.25059 


Eq. 133. 






c=1780.7 


c=1780.7 


c=1780.7 


c=1780.7 


ri = 1912.4.5 


r2= 1907.75 


ri = 1912.45 


r2=1907.75 


r4= 1430.35 


r4= 1430.35 


r3= 1435.05 


r3= 1435.05 


2|5123.50 


2|5118.80 


2|5128.20 


2|5123.50 


51 = 2561.75 


52 = 2559.40 


53 = 2564.10 


54 = 2561.75 


«i-ri= 649.30 


52-r2= 651.65 


a3-rj= 651.65 


54-r2= 654.00 


»i-r4=1131.40 


-r4- 1129.05 


53-r3 = 1129.05 


54-r3== 1126.70 



log 2 = 0.30103 
(5i-ri); log 649.30 = 2.81244 
{si-n)\ log 1131.40 = 3.05361 



ri = 1912.45; 


log = 3. 28159; 


co-log = 6. 71841 


r4= 1430.35; 


log = 3.15544; 


co-log = 6 . 84456 


Fi=62° 25' 31 


Ll 


log vers 62° 25' 31" = 9. 73006 






log 2 = 0.30103 






(«2 ~ r-z) ; log 651 . 65 = 2 . 81401 






(s2-r4); log 1129.05 = 3.05271 


r2= 1907.75; 


log = 3. 28052; 


co-log = 6. 7 1948 


r4= 1430.35; 


log = 3. 15544; 


co-log = 6. 84456 


F, = 62° 33' 55 


li 


log vers 62^^ 33' 55" = 9. 73180 



§ 279. 



SWITCHES AND CROSSINGS. 



323 



ri = 1912.45: log = 3.28159; 
r3= 1435.05; log = 3. 15686; 
Fa = 62^ 2r b7"% 



r2=1907.75; log = 3. 28052; 
ra = 1435 . 05 ; log = 3 . 15686 ; 



log 2 = 0.30103 

(«3-ri); log 651.65 = 2.81401 

(ss-ra); log 1129.05 = 3.05271 

co-log = 6. 7 1841 

co-log = 6_84313 

log vers 62° 21 ^ 57^^ = 9.72930 

log 2 = 0.30103 

(«4-r2); log 654.00 = 2.81558 

(«4-7-3); log 1126.70 = 3.05181 

co-log = 6.71948 

co-log = 6. 843 13 

log vers 62° 30' 14'^ = 9. 73103 



^4 = 62° 30' 14'' ; 

As a check, the mean of the frog angles = 62° 27' 54 ', which is within 6" of 
the value of M, 



Eq. 134. 

(7iC2F4 = 45°37' 51" i 



log c = 3. 25059] 



CiC2F2 = 45° 28' 17"', 

J^C^=45° 37' 51"-45» 28' 17"== 0°0y 34" . 



log sin F4 = 9.94794 

log r3 = 3. 15686 

co-log c = 6. 74940 

sin CiC2F4 = 9. 85421 

log sin ^2 = 9. 948 18 

log r4 = 3.15544 

co-log c = 6. 74940 

sinCiC2F2 = 9.85303 



log 2 = 0.30103 
log r2 = 3. 28052 

i(0° 09' 34" ) = 0° 04' 47" ; log sin = (| • ^p|| 
log ^2^4 = 0.72500 

sin Fi = 9. 94 763 

log ri = 3. 281 59 

co-log c = 6. 74940 

sin FiCiCo = 9 . 97863 

sin ^2 = 9.94818 

log r2 = 3. 28052 

co-log c = 6. 74940 

sinF2CiC2=9.97811 

log 2 = 0.30103 
log r4 = 3. 15544 

1(0° 12' 44") = 0° 06' 22"; log sin= Z^. 68557 

V 2. 58206 
FiFa = 5.298; logFiF2 = 0.72411 

As a check, F2-P4 and FyF2 are very nearly equal, as they should 
be. 



^2^4 = 5.309 ; 
Eq. 135. 

FiCiC2=72° IC 22"5 



F2CiC2 = 71° 57' 38"; 

FiCiF2 = 72° 10' 22" -71° 57' 38" -0° 12' 44'^ 



324 RAILROAD CONSTRUCTION. § 279a. 

279a. Slips. In a crowded yard the possible number of 
track movements from one track to another may be greatly in- 
creased and even multiplied by the adoption of ^' slips/' such 
as are illustrated in Fig. 159a, which shows a '^ single slip^' 
and also a *' double slip.'' In one case the crossing of two rails 
is accomplished by using fxed ''frogs/' although it should be 
realized that these frogs are different from an ordinary switch 
frog. A comparison of the continuity of the running rails 
through these frogs and through ordinary frogs, such as are 
illustrated in Fig. 130, or Plate VIII, will show the difference. 
In the case of the double slip the frogs are movable. Either 
fixed or movable frogs may be used for either single or double 
slips. As shown in the figure, the levers are so connected that 
the several operations necessary to set the rails for any desired 
train movement are accomplished by one motion. These slips 
can be used for frog angles varying from No. G to No. 15. 



§ 2790. 



SWITCHES AND CROSSINGS. 



325 




pTrt 1 /;Q/t 



CHAPTER XII. 

MISCELLANEOUS STRUCTURES A.ND BUILDINGS. 

WATER-STATIONS AND WATER-SUPPLY. 

280. Location. The water-tank on the tender of a locomo- 
tive has a capacity of from 3000 to 7000 gallons — sometimes less, 
rarely very much more. The consumption of water is very vari- 
able, and will correspond very closely with the work done by 
the engine. On a long down grade it is very small; on a ruling 
grade going up it may amount to 150 gallons per mile in ex- 
ceptional cases, although 60 to 100 gallons would be a more usual 
figure. A passenger locomotive can run 60 miles or more 
on one tankful, but freight work requires a shorter interval 
between water-stations. On roads of the smallest traffic, 
15 to 20 miles should be the maximum interval between stations; 
10 miles is a more common interval on heavy traffic-roads. But 
these intervals are varied according to circumstances. In the 
early history of some of the Pacific railroads it was necessary to 
attach one or more tank-cars to each train in order to maintain 
the supply for the engine over stretches of 100 miles and over 
where there was no water. Since then water-stations have been 
obtained at great expense by boring artesian wells. The indi- 
vidual locations depend largely on the facility with which a suffi- 
cient supply of suitable water may be obtained. Streams inter- 
secting the railroad are sometimes utilized, but if such a stream 
passes through a limestone region the water is apt to be too hard 
for use in the boilers. More frequently wells are dug or bored. 
\Mien the local supply at some determined point is unsuitable, 
and yet it is necessary to locate a water-station there, it may 
be found justifiable to pipe the water several miles. The con- 
struction of municipal water-works at suitable places along the 
line has led to the frequent utilization of such supplies. In such 
cases the railroad is generally the largest single consumer and 
obtains the most favorable rates. When possible, water-stations 
are located at regular stopping points and at division termini. 

326 



§ 281. MISCELLANEOUS STRUCTURES AND BUILDINGS. 327 

281. Required qualities of water. Chemically pure water is 
unknown except as a laboratory product. The water supplied 
by wells, springs, etc., is alwa3's more or less charged with cal- 
cium and magnesium carbonates and sulphates, as well as other 
impurities. The evaporation of water in a boiler precipitates 
these impurities to the lower surfaces of the boiler, where they 
sometimes become incrusted and are difficult to remove. The 
protection of the iron or steel of a boiler from the fierce heat of 
the fire depends on the presence of water on the other side of the 
surface, which will absorb the heat and prevent the metal from 
assuming an excessively high temperature. If the water side 
of the metal becomes covered or incrusted with a deposit 
of chemicals, the conduction of heat to the water is much less 
free, the metal will become more heated and its deterioration or 
destruction will be much more rapid. An especially common 
effect is the production of leaks around the joints between tubes 
and tube-sheets and the joints in the boiler-plates. Such in- 
jury can only be prevented by the application of one (or both) 
of two general methods — (a) the frequent cleaning of the boilers 
and (6) the chemical purification of the water before its intro- 
duction into the boiler. Although ^'manholes" and ^'hand- 
holes" are made in boilers, it is physically impossible to clean 
out every corner of the inside of a boiler where deposits will form 
and where they are especially objectionable — on the tube-sheets. 
Such a cleaning is troublesome and expensive. 

Chemical purification is generally accomplished by treating 
the water before it enters the boiler. The reagents chiefly em- 
ployed are quicklime and sodium carbonate. Lime precipi- 
tates the bicarbonate of lime and magnesia. Sodium carbo- 
nate gives, by double decomposition in the presence of sulphate 
of lime, carbonate of lime, which precipitates, and soluble sul- 
phate of soda, which is non-incrustant. When this is done in a 
purifying tank, the purified water is drawn off from the top of 
the tank and supplied pure to the engines. The precipitants are 
drawn off from the settling-basin at the bottom of the tank. 
This purification, which makes no pretense of being chemically 
perfect, may be accomplished for a few cents per 1000 gallons. 
It is used much more extensively in Europe than in this countr}^, 
the Southern Pacific being the only railroad which has employed 
such methods on a large scale. Reliance is frequently placed 
on the employment of a ^' non-incrustant" which is introduced 



328 



RAILROAD CONSTRUCTION. 



§281. 



directly into the boiler. When no incrustation takes place 
the accumulation of precipitant and mud in the bottom of the 
boiler may be largely removed by mere ''blowing off" or by 
washing out with a hose. 

On the other hand, there is the exceptional case that the 
water may be too pure. It is well known that distilled water 
has a very strong corrosive action on iron and that it is possible 
for the water to be so pure that corrosion of the boiler tubes 
will be accelerated and that the boilers will rapidly deteriorate 
in this way. It is therefore occasionally necessary to add a 
small portion of lime to a very sot; water, so that a very thin 
scale will form over the surface of the iron, which will protect 
the iron from corrosion. 

American practice may therefore be summarized as follows: 
(a) Employing as pure water as possible; (b) cleaning out boil- 
ers by ''blowing off" or by washing out with a hose or by physi- 
cal scraping at more or less frequent intervals or when other 
repairs are being made; (c) the occasional employment of non- 
incrustants; (d) the occasional chemical treatment of water 
before it enters the tender-tank. 

282. Tanks. Whatever the source, the water must be led 
or pumped into tanks which are supported on frames so that the 

bottoms of the tanks are 
about 12 feet above the 
rails. Wooden tanks hav- 
ing a diameter of 24 feet, 
16 feet high, and with a 
capacity of over 50,000 
gallons, are frequently 
employed. Iron or steel 
tanks are also used. 

In Table XIV is shown 
the capacity of cylindrical 
water-tanks in United 
States standard gallons of 
231 cubic inches. From 
this table the dimen- 
sions of a tank of any 
desired capacity may 
readily be found. Two or more tanks are sometimes used 
rather than construct one of excessive size. The smaller sizes 




FiQ. 160. — Water-tank. 



§ 282. MISCELLANEOUS STRUCTURES AND BUILDINGS. 329 

TABLE XIV. CAPACITY OF CYLINDRICAL WATER-TANKS IN 

UNITED STATES STANDARD GALLONS OF 231 CUBIC INCHES. 



Height 






Diameter of tank in feet. 






feet. 


10 


12 


14 


16 


18 


20 


22 


24 


6 

7 

8 

9 

10 


3525 
4113 
4700 
5288 
5875 


5076 
5922 
6768 
7614 
8460 


6909 

8061 

9212 

10364 

11515 


9024 
10528 
12032 
13536 
15041 


11421 
13325 
15229 
17132 
19036 


14101 
16451 
18801 
21151 
23501 


17062 
19905 
22749 
25592 
28436 


20305 
23689 
27073 
30457 
33841 


11 
12 
13 
14 
15 


6463 
7050 
7638 
8225 
8813 


9306 
10152 
10998 
11844 
12690 


12667 
13819 
14970 
16122 
17273 


16545 
18049 
19553 
21057 
22561 


20939 
22843 
24746 
26650 
28554 


25851 
28201 
30551 
32901 
35251 


31280 
34123 
36967 
39810 
42654 


37225 
40609 
43994 
47378 
50762 


16 
17 
18 
19 
20 


9400 

9988 

10575 

11163 

11750 


13536 
14383 
15229 
16075 
16921 


18425 
19576 
20728 
21879 
23031 


24065 
25569 
27073 
28577 
30081 


30457 
32361 
34264 
36168 
38071 


37601 
39951 
42301 
44652 
47002 


45498 
48341 
51185 
54028 
56872 


54146 
57530 
60914 
64298 
67682 


21 
22 
23 
24 
25 


12338 
12925 
13513 
14101 
14688 


17767 
18613 
19459 
20305 
21151 


24182 
25334 
26485 
27637 
28789 


31.585 
33089 
34593 
36097 
37601 


39975 
41879 
43782 
45686 
47589 


49352 
51702 
54052 
56402 
58752 


59716 
62559 
65403 
68246 
71090 


71067 
74451 
77835 
81219 
84603 



shown in the table are of course too small for ordinary use, 
but that part of the table was filled out for its possible con- 
venience otherwise. On single-track roads where all engines 
use one track the tank may be placed 8' 5" from the track 
center; this gives sufficient clearance and yet permits the use 
of a single swinging pipe which will reach from the bottom 
of the tank to the tender manhole. In I^ig» 160 is illustrated 
one form of wooden tank. They are preferably manufactured 
by those who make a special business of it and who by the use 
of special machinery can insure tight joints. When it is incon- 
venient to place the tank near the track, or when there is a 
double track, a 'stand-pipe" becomes necessary. See §285. 
One of the most difficult and troublesome problems is to prevent 
freezing, particularly in the valves and pipes Not only are the 
pipes carefully covered but fires must be maintained during cold 
weather. When the pumping is accompHshed by means of a 
steam-pump, suppHed from a steam-boiler in the pump -house 
under the tank, coils of steam-pipe may be employed to heat the 
water or to heat the pipes Partial protection may be obtained 
by means of a double roof and double bottom, the spaces being 
filled with sawdust or some other non-conductor of heat. 



330 RAILROAD CONSTRUCTION. § 283. 

283. Pumping. The pumping is done most reliably wdth 
steam-pumps or gas-engines, although hot-air engines,^windmills, 
and even man-power are occasionally employed. Economy of 
operation requires that the water-stations shall be so located 
that each tank shall be used regularly and that each pump shall 
be regularly operated for maintaining the water-supply. On 
the other hand, the pump should not be required to regularly 
work at night to maintain the supply and should have an excess 
capacity of say 25%. When a tank is but little used, it will still 
require the labor of an attendant, and his time will be largely 
wasted unless he can be utilized for other labor about the station. 
In recent years gasoline has been extensively employed as a fuel 
for the pumping-engines. The chief advantages of its use lies in 
the extreme simplicity of the mechanism and the very slight 
attention it requires, Avhich permits their being operated by 
station-agents and others, who are paid $10 per month extra, 
instead of paying a regular pumper $35 per month. ^' Screen- 
ings," ''slack coal," etc., are used as fuel for steam-pumps and 
may frequently be delivered at the pump-house at a cost not 
exceeding 30 cents per ton, but even at that price the cost of 
pumping per thousand gallons, although dependent on the hori- 
zontal and vertical distances to the source of supply and to 
the tank, will generally run at 2 cents to 6 cents per 1000 gallons. 
In many cases where steam plants have been replaced by gasoline 
plants, the cost of pumping per 1000 gallons has been reduced 
to one third or even one fourth of the cost of steam pumping. 
Of course the cost, using windmills, is reduced to the mere 
maintenance of the machiner}^, but the unreliability of wind as 
a motive power and the possibiHty of its failure to supply water 
when it is imperatively needed has made this form of motive 
power unpopular. (See report to Ninth Annual Convention 
of the Association of Railway Superintendents of Bridges and 
Buildings, Oct. 1899.) 

284. Track tanks. These are chiefly required as one of the 
means of avoiding delays during fast-train serA'ice. A trough, 
made of steel plate, is placed between the rails on a stretch of 
perfectly level track. A scoop on the end of a pipe is lo^Acred 
from under the tender into the tank while the train is in motion. 
The rapid motion scoops up the water, which then flows mto the 
tender tank. The following brief description of an 'nstallation 
on the Baltimore & Ohio Railroad between Baltimore and 



§ 286. MISCELLANEOUS STRUCTURES AND BUILDINGS. 331 

Philadelphia will answer as a general description of the 
method. The trough is made of ^V' steel plate, 19'' wide, 6" 
deep, and has a length of 1200 feet. There is riveted on each 
side a line of H'' X2" Xi" angle bars. These angle bars rest on 
the ties. Ordinary track spikes hold these angle bars to the 
ties, but permit expansion as with rails. The tanks are firmly 
anchored at the center, the ends being free to expand or con- 
tract. The plates are 15 feet long and are riveted with jV' 
rivets, 20 rivets per joint. At each end is an inclined plane 
13' 8" long. If the fireman should neglect to raise the scoop 
before the end of the tank is reached, the inclined plane will 
raise it automatically and a catch wiU hold it raised. AVater 
is supplied to the tanks by a No. 9 Blake pump having a 
capacity of 260 gallons per minute. During cold weather, 
freezing is prevented by injecting into the side of the tanks, 
at intervals of 45 feet, jets of steam, which come through 
J" holes. Two boilers of 80 and 95 H.P. are required for pump- 
ing and to keep the water from freezing. During warm 
weather an upright 25 H.P. boiler suffices for the pumping. 
The cost of installation was about $10,000 to $11,000, the cost of 
maintenance being about $132.50 per month. 

285. Stand-pipes. These are usually manufactured by those 
who make a specialty of such track accessories, and who can 
ordinarih^ be trusted to furnish a correctly designed article. In 
Fig. 161 is shown a form manufactured by the Sheffield Car Co. 
Attention is called to the position of the valve and to the device 
for holding the arm parallel to the track when not in use so that 
it will not be struck by a passing train. When a stand pipe is 
located between parallel tracks, the strict requirements of clear- 
ance demand that the tracks shall be bowed outward slightly. 
If the tracks were originally straight, they may be shoved over by 
the trackmen, the shifting gradually running out at about 100 
feet each side of the stand-pipe. If the tracks were originally 
curved, a slight change in radius will suffice to give the necessary 
extra distance between the tracks. 

BUILDINGS. 

286. Station platforms. These are most commonly made of 
planks at minor stations. Concrete is used in better-class work, 
also paving brick. An estimate of the cost of a platform of paving 
brick laid at Topeka, Kan., was $4.89 per 100 square feet when 



332 



RAILROAD CONSTRUCTION. 



§ 286. 



laid flat and $7.24 per 100 square feet when laid on edge. The 
curbing cost 36 cents per linear foot. Cinders, curbed by timbers 




Fig. 161. — Stand-pipe. 
or stone, bound by iron rods, make a cheap and fairly durable 
platform, but in wet weather the cinders T\all be tracked into 



§ 287. MISCELLANEOUS STRUCTURES AND BUILDINGS. 333 

the stations and cars. Three inches of crushed stone on a 
cinder foundation is considered to be still better, after it is once 
thoroughly packed, than a cinder surface. 

Elevation. — The elevation of the platform with respect to 
the rail has long been a fruitful source of discussion. Some roads 
make the platforms on a level with the top of the rail, others 
3" above, others still higher. As a matter of convenience to 
the passengers, the majority find it easier to enter the car from 
a high platform, but experience proves that accidents are more 
numerous with the higher platforms, unless steps are discarded 
altogether and the cars are entered from level platforms, as is 
done on elevated roads. As a railroad must generally pay 
damages to the stumbling passenger, they prefer to build the 
lower platform. Convenience requires that the rise from the 
platform to the lowest step should not be greater than the rise 
of the car steps. This rise is variable, but with the figures usually 
employed the application of the rule will make the platform 
5'' to 15" above the rail. 

Position with respect to tracks. — Low platforms are gen- 
erally built to the ends of the ties, or, if at the level of the top 
of the rail, are built to the rail head. Car steps usually 
extend 4' Q" from the track center and are 14" to 24" above the 
rail. The platform must have plenty of clearance, and when 
the platform is high its edge is generally required to be 5' 6" 
from the track center. 

287. Minor stations. For a complete discussion of the design 
of stations of all kinds, including the details, the student is re- 
ferred to '^Buildings and Structures of American Railroads," 
by Walter G. Berg, now Chief Engineer of the Lehigh Valley 
Railroad. The subject is too large for adequate discussion here, 
but a few fundamental principles will be referred to. 

Rooms required. An office and waiting-room is the mini- 
mum. A baggage-room, toilet-rooms, and express office are 
successively added as the business increases. In the Southern 
States a separate waiting-room for colored people is generally 
provided. It used to be common to have separate waiting-rooms 
for men and women. Experience proved that the men's wait- 
ing-room became a lounging place and smoking-room for loafers, 
and now large single waiting-rooms are more common even in 
the more pretentious designs, smoking being excluded. The 
office usually has a bay window, so that a more extended view 



334 RAILROAD CONSTRUCTION. § 287. , 

of the track is obtainable. The women's toilet-room is entered 
from the waiting-room. The men's toilet-room, although built 
immediately adjoining the other in order to simplify the plumb- 
ing, is entered from outdoors. Old-fashioned designs built the 
station as a residence for the station-agent; later designs have 
very generally abandoned this idea. ''Combination" stations 
(passenger and freight) are frequently built for small local 
stations, but their use seems to be decreasing and there is now a 
tendency to handle the freight business in a separate building. 

288. Section-houses. These are houses built along the right- 
of-way by the railroad company as residences for the trackmen. 
The liability of a wreck or washout at any time and at any part 
of the road, as well as the convenience of these houses for ordinary 
track labor, makes it all but essential that the trackmen should 
live on the right-of-way of the road, so that they may be easily 
called on for emergency service at any time of day or night. 
This is especially true when the road passes through a thinly 
settled section, where it would be difficult if not impossible to 
obtain suitable boarding-places. It is in no sense an extrava- 
gance for a railroad to build such houses. Even from the direct 
financial standpoint the expense is compensated by the corre- 
sponding reduction in wages, which are thus paid partly in free 
house rent. And the value of having men on hand for emergen- 
cies will often repay the cost in a single night. Where the coun- 
try is thickly settled the need for such houses is not so great, and 
railroads will utilize or perhaps build any sort of suitable house, 
but on Southern or Western roads, where the need for such 
houses is greater, standard plans have been studied with great 
care, so as to obtain a maximum of durability, usefulness, com- 
fort, and economy of construction. (See Berg's Buildings, etc., 
noted above.) On Northwestern roads, protection against cold 
and rain or snow is the chief characteristic; on Southern roads 
good ventilation and durability must be chiefly considered. 
Such houses may be divided into two general classes — (a) those 
which are intended for trackmen only and which may be built 
with great simplicity, the only essential requirements being a 
living-room and a dormitory, and (6) those which are intended 
for families, the houses being then distinguished as ''dwelling- 
houses for employees. 

289. Engine-houses. Small engine-houses are usually built 
rectangular in plan. Their minimum length should be some- 



§ 289. MISCELLANEOUS STRUCTURES AND BUILDINGS. 335 

what greater than that of the longest engine on the road. They 
may be built to accommodate two engines on one track, but 
then they should be arranged to be entered at either end, so that 
neither engine must wait for the other. In width there may be 
as many tracks as desired, but if the demand for stalls is large, 
it will probably be preferable to build a *' roundhouse.'' Rect- 
angular engine-houses are usually entered by a series of parallel 
tracks switching off from one or more main tracks, no turn-table 
being necessary. If a turn-table is placed outside (because one 



% cement « » , 

4 X 12-9i drift bolts 




Fig. 162. — Engine-house. 

is needed at that part of the road) enough track should be allowed 
between the house and the turn-table so that engines may be 
quickly removed from the engine-house in case of fire without 
depending on the turn-table to get them out of danger. 

Roundhouses. The plan of these is generally polygonal 
rather than circular. The straight walls are easier to build; the 
construction is more simple, and the general purpose is equally 
well served. They may be built as a part of a circle or a com- 
plete circle, a passageway being allowed, so that there are two 
entrances instead of one. When space is very limited a round- 
house with turn-table will accommodate more engines in pro- 
portion to the space required (including the approaches) than a 
rectangular house. The enlarged space on the outer side of each 
segment of a roundhouse furnishes the extra space which is needed 
for the minor repairs which are usually made in a roundhouse. 
One disadvantage is that supervision is not quite so easy or effec- 



336 RAILROAD CONSTRUCTION. § 289. 

tive as in rectangular houses. Of course such houses are used 
not only for storing and cleaning engines, but also for minor 
repairs which do not require the engine to be sent to the shops 
for a general overhauling. 

Construction. The outer walls are usually of brick. The 
inner walls consist almost entirely of doors and the piers between 
them, although there is usually a low wall from the top of the 
door frames to the roof line, which usually slopes outward so as 
to turn rain-water away from the central space. 

Roofs. Many roofs have been built of slate with iron truss 
framing, with the idea of maximum durabihty. The slate is good, 
but experience shows that the iron framing deteriorates very 
rapidly from the action of the gases of combustion of the engines 
which must be '^ fired'' in the houses before starting. Roof 
frames are therefore preferably made of wood. 

Floors. These are variously constructed of cinders, wood, 
brick, and concrete. Brick has been found to' be the best ma- 
terial! Anything short of brick is a poor economy; concrete is 
very good if properly done but is somewhat needlessly expensive. 

Ventilation. This is a troublesome and expensive matter. 
The general plan is to have ''smoke-jacks" which drop down 
over the stack of each engine as it reaches its precise place in its 
stall and which will carry away all smoke and gas. Such a 
movable stack is most easily constructed of thin metal— say 
galvanized iron— but these will be corroded by the gases of 
combustion in two or three years. Vitrified pipe, cast iron, 
expanded metal and cement, and even plain wood painted with 
''fireproof' paint, have been variously tried, but all methods 
have their unsatisfactory features. (For an extended discus- 
sion of roundhouse floors and ventilation see the Proc. Assoc. 
of Railway Supts. of Bridges and Buildings for 1898, pp. 112-135.) 

SNOW STRUCTURES. 

290. Snow-fences. Snow structures are of two distinct 
kinds— fences and sheds. A snow-fence impHes drifting snow- 
snow carried by w^nd— and aims to cause all drifting snow to be 
deposited away from the track. Some designs actually succeed 
in making the wind an agent for clearing snow from the track 
where it has naturally fallen. A snow-fence is placed at right 
angles to the prevaiHng direction of the wind and 50 to 100 feet 
awa^r from the tracks. WTien the road line is at right angles to 



§ 291. MISCELLANEOUS STRUCTURES AND BUILDINGS. 337 

the prevailing wind, the right-of-way fence may be built as a 
snow-fence — high and with tight boarding. Hedges have some- 
times been planted to serve this purpose. When the prevailing 
wind is oblique, the snow fences must be built in sections where 
they wdll serve the best purpose. The fences act as wind break- 
ers, suddenly low^ering the velocity of the wind and causing the 
snow carried by the wind to be deposited along the fence. 
Portable fences are frequently used, which are placed (by per- 
mission of the adjoining property owners) outside of the right- 
of-way. If a drift forms to the height of the portable fence the 
fence may be replaced on the top of the drift, where it may act 
as before, forming a still higher drift. When the prevailing 
wind runs along the track line, snow-fences built in short sec- 
tions on the sides will cause snow to deposit around them 
while it scours its way along the track line, actually clearing 
it. Such a method is in successful operation at some places on 
the White Mountain and Concord divisions of the Boston & 
Maine Railroad. Snow-fences, in connection with a moderate 
amount of shoveling and plowing, suffice to keep the tracks 
clear on railroads not troubled with avalanches. In such cases 
snow^-sheds are the only alternative. 

291. Snow-sheds. These are structures which will actually 
keep the tracks clear from snow regardless of its depth outside. 
Fortunately they are only necessary in the comparatively rare 
situations where the snowfall is excessive and where the snow 
is liable to slide down steep mountain slopes in avalanches. 
These avalanches frequently bring down with them rocks, trees, 
and earth, which would otherwise choke up the road-bed and 
render it in a moment utterly impassable for weeks to come. 
The sheds are usually built of 12"Xl2" timber framed in about 
the same manner as trestle timbering; the ^^ bents" are some- 
times placed as close as 5 feet, and even this has proved insuffi- 
cient to withstand the force of avalanches. The sheds are there- 
fore so designed that the avalanche will be deflected over them 
instead of spending its force against them. Although these 
sheds are only used in especially exposed places, yet their length 
is frequently very great and they are liable to destruction by fire. 
To confine such a fire to a limited section, '^fire-breaks" are 
made — i.e., the shed is discontinued for a length of perhaps 100 
feet. Then, to protect that section of track, a V-shaped de- 
flector will be placed on the uphill side which wdll deflect all 



338 



RAILROAD CONSTRUCTION. 



§291. 



descending material so that it passes over the sheds. Solid crib 
work is largely used for these structures. Fortunately suitable 
timber for such construction is usually plentiful and cheap 
where these structures are necessary. Sufficient ventilation 
is obtained by longitudinal openings along one side immediately 
under the roof. ''Summer'' tracks are usually built outside 
the sheds to avoid the discomfort of passing through these semi- 
tunnels in pleasant weather. The fundamental elements in 
the design of such structures is shown in Fig. 163, which illus- 
trates some of the sheds used on the Canadian Pacific Railroad 




Lever-.faJI sb.ed 



Fig. 163. — Snow-sheds — Canadian Pacific Railroad, 



292. Turn-tables. The essential feature of a turn-table is a 
carriage of sufficient size and strength to carry a locomotive, 
the carriage turning on a pivot of sufficient size to carry such a 
load. The carriage revolves in a circular pit w^hose top has 
the same general level as the surrounding tracks. The car- 
riages were formerly made largely of wood; very many of 
those still in use are of cast iron. Structural steel is now uni- 



§ 292. MISCELLANEOUS STRUCTURES AND BUILDINGS. 339 

versally employed for all modern work and since the construc- 
tion of the carriage and the pivot is a special problem in struc- 
tures, no further attention will here be paid to the subject, 
except to that part which the railroad engineer must work out 
— laying out the site and preparing the foundation. The 
minimum length of such a carriage (and therefore the diameter 
of the pit) is evidently the length over all of the longest engine 
and tender in use on the road. Usually 60-foot turn-tables 
will suffice for an ordinary road, and for light-traffic roads 
employing small engines, 50 feet or even less may be sufficient. 
Many of the heavier freight engines of recent make have a total 
length of about 65 feet; therefore 70-foot turn-tables are a 
better standard for heavy-traffic roads. A retaining-wall 
should be built around the pit. The stability of this wall imme- 
diately under the tracks should be especially considered. The 
most important feature is the stability of the foundation of the 
pivot, which must sustain a concentrated pressure, more or less 
eccentric, of perhaps 150 tons. When firm soil or rock may 
be easily reached, this need give no trouble, but in a soft, treach- 
erous soil a foundation of concrete or piling may be necessary. 
If the soil is very porous, it may be depended on to carry away 
all rain-water which may fall into the pit before the foundations 
are affected, but when the soil is tenacious it may be necessary 
to drain the subsoil thoroughly and carry off immediately all 
surface drainage by means of subsoil pipes which have a suit- 
able outfall. 

The location of the turn-table in the yard is a part of the 
general subject of ^' Yards," and will be considered in the next 
chapter. 



CHAPTER XIII. 

YARDS AND TERMINALS. 

293. Value of proper design. A large part of the total cost of 
handling traffic, particularly freight, is that incurred at terminals 
and stations. In illustration of this, consider the relative total 
cost of handling a car-load of coal and a car-load (of equal 
weight) of mixed merchandise. The coal will be lokded in 
bulk on the cars at the mines, where land is comparatively 
cheap, and the cars grouped into a train without regard to order, 
since they are (usually) uniform in structure, loading, and con- 
tents. When the terminal or local station is reached they are 
run on tracks occupying property w^hich is usually much cheaper 
than the site of the terminal tracks and freight-houses; they are 
unloaded by gravity into pockets or machine conveyors and the 
empty cars are rapidly hauled by the train-load out of the way. 
On the other hand, the merchandise is loaded by hand on the car 
from a freight-house occupying a central and valuable location, 
the car is hauled out into a yard occupying valuable ground, is 
drilled over the j^ard tracks for a considerable aggregate mileage 
before starting for its destination, w^here the same process is re- 
peated in inverse order. In either case the terminal expenses are 
evidently a large percentage of the total cost and, once loaded, 
it makes but little difference just how far the car is hauled to the 
other terminal. But the very evident increase in terminal charges 
for general merchandise over those for coal (large as they are) 
gives a better idea of the magnitude of terminal charges. 

Many yards are the result of growth, adding a few tracks at a 
time, without much evidence of any original plan. In such 
cases the yard is apt to be very inefficient, requiring a much 
larger aggregate of drilling to accomplish desired results, requir- 
ing much more time and hence blocking traffic and finally adding 
greatly to the cost of terminal service, although the fact of its 
being a needless addition to cost may be unsuspected or not fully 
appreciated. An unwillingness or inability to spend money for 

340 



J 295. YARDS AND TERMINALS. 341 

the necessary changes, and the difficulty of making the changes 
while the yard is being used, only prolong the bad state of 
affairs and an inefficient makeshift is frequently adopted. As- 
sume that an improvement in the design of the yard will permit 
a saving of the use of one switching engine, or for example, that 
the work may be accomplished with three switching engines in- 
stead of four. Assuming a daily cost of $25, we have in 313 
working days an annual saving of $7825, which, capitalized at 
5%, gives $156,500, enough to reconstruct any ordinary yard.* 

294. Divisions of the subject. The subject naturally divides 
itself into three heads — (a) Yards for receiving, classifying, and 
distributing freight cars, called more briefly freight yards; (h) 
yards and conveniences for the care of engines, such as ash tracks, 
turn-tables, coal-chutes, sand-houses, water-tanks, or water 
stand-pipes, etc., and (c) passenger terminals. 

FREIGHT YARDS. 

295. General principles. It should be recognized at the start 
that at many places an ideally perfect yard is impossible, or at 
least impracticable, generally because ground of the required 
shape or area is practically unobtainable. But there are some 
general principles which may and should be follow^ed in every yard 
and other ideals which should be approached as nearly as possi- 
ble. Nevertheless every yard is an independent problem. Be- 
fore taking up the design of freight yards, it is first necessary to 
consider the general object of such yards and the general princi- 
ples by which the object is accomplished. These may be briefly 
stated as follows: 

1. A yard is a device, a machine, by w^hich incoming cars are 
sorted and classified — some sent to warehouses for unloading, 
some sent to connecting railroads, some made up for local dis- 
tribution along the road, some sent for repairs, and, in short a 
device by which all cars are sent through and out of the yard as 
quickly as possible. 

2. Except when a road's business is decreasing, or when its 
equipment is greater than its needs and its cars must be stored, 
efficiency of management is indicated by the rapidity with which 
the passage of cars through the yard is accomplished. 

3. When a yard is the terminal of a ''division," the freight 

* Estimate of Mr. H. G. Hetzler, C, B. & Q. Ry. 



342 RAILROAD CONSTRUCTION. § 295. 

trains wall be pulled into a ''receiving track" and the engine and 
caboose detached. The caboose will be run on to a ''caboose 
track," which should be conveniently near, and the engine is run 
off to the engine yard. If the train is a " through" train and no 
change is to be made in its make-up, it will only need to wait for 
another engine and perhaps another caboose. If the cars are to 
be distributed, they will be drawn off by a switching engine to 
the "classification yard." 

4. The design of a yard is best studied by first picking out the 
ladder tracks and the through tracks which lead from one divi- 
sion of the yard to another. These are tracks which must always 
be kept open for the passage of trains, in contradistinction to 
the tracks on which cars may be left standing, even though it is 
only for a few moments, while drilling is being done. Such a set 
of tracks, which may be called the skeleton of the yard, is shown 
by heavy lines in Fig. 164. Each line indicates a pair of rails. 
The tracks of the storage yards are shown by the lighter lines. 

5. There is a distinct advantage in having all storage tracks 
double-ended — except "team tracks." Team tracks are those 
which have spaces for the accommodation of teams, so that load- 
ing or unloading may be done directly between the cars and teams. 
To avoid the necessity of teams passing over the tracks, these are 
best placed on the outskirts of the yard and consist of short stub- 
sidings arranged in pairs. But storage tracks should have an 
outlet at each end so as to reduce the amount of drilling neces 
sary to reach a car which may be at the extreme end of a long 
string of cars. This is done usually by means of two "ladder" 
tracks, parallel to each other, which thus make the storage 
tracks between them of equal length. 

6. The equality of length of these storage tracks is a point in- 
sisted on by many, but on the other hand, trains are not always of 
uniform length even on any one division. Loaded trains and 
trains of empties Tvdll vary greatly in length, and the various 
styles and weights of freight engines employed necessitate other 
variations in the weights and lengths of trains hauled. With 
storage tracks of somewhat variable length a larger percentage 
of track length may be utilized, there will be less hauHng over a 
useless length of track, and (assuming that the plot of ground 
available for yard purposes has equally favorable conditions for 
yard design) more business may be handled in a yard of given 
area. 



? 295. 



YARDS AND TERMINALS. 



343 




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344 RAILROAD CONSTRUCTION. § 295. 

7 Yards are preferably built so that the tracks have a grade 
of 0.5% — sometimes a little more than this — in the direction of 
the traffic through the yard. This grade, which will overcome a 
tractive resistance of 10 pounds per ton, will permit cars to be 
started down the ladder tracks by a mere push from the switch- 
ing engine. They are then switched on to the desired storage 
track and run down that track by gravity until stopped at the 
desired place by a brakeman riding on the cars 

8. Although not absolutely necessar}% there is an advantage 
in having all frog numbers and switch dimensions uniform. 
No. 7 frogs are most commonly used. Sharper- angled frogs 
make easier riding, less resistance and less chance of derailment, 
but on the other hand require longer leads and more space. No. 
6 and even No. 5 frogs are sometimes used on account of economy 
of space, but they have the disadvantages of greater tractive re- 
sistance, greater wear and tear on track and rolling stock, and 
greater danger of derailment. 

296. Relation of yard to main tracks. Safety requires that 
there should be no connection between the yard tracks and the 
main tracks except at each end of the yard, where the switches 
should be amply protected by signals. Sometimes the main 
tracks run through the yard, making practically two yards— -one 
for the traffic in either direction — but this either requires a double 
layout of tracks and houses (such as ash tracks, coal-chutes, sand- 
houses, etc.), or a very objectionable amount of crossing of the 
main-line tracks. The preferable method is to have the main hne 
tracks entirely on the outside of the yard. A method which is in 
one respect still better is to spread the main tracks so that they 
run on each side of the yard. In this case there is never any 
necessity to cross one main track to pass from the yard to the 
other main track; a train may pass from the yard to either 
main track and still leave the other main track free and open. 
The ideal arrangement is that by which some of the tracks cross 
over or under all opposing tracks. By this means a.11 connections 
between the j^ard and the main tracks maybe by ''trailing" 
s\vitches; that is, trains will run on to the main track in the 
direction of motion on that main track. Of course all this 
applies only to double main track. 

An important element of yard design is to have a few tracks im- 
mediately adjoining the main tracks and separate from the yard 
proper on which outgoing trains may await their orders to take 



§ 296. 



YARDS AND TERMINALS. 



345 




346 RAILROAD CONSTRUCTION. § 297. 

the main track. When the orders come, they may start at once 
without an}^ delay, without interfering with any j^ard operations, 
and they are not occupying tracks which may form part of the 
system needed for switching. 

297. Minor freight yards. The term here refers to the sub- 
stations, only found in the largest cities, to which cars will be sent 
to save in the amount of necessary team hauling and also to re- 
lieve a congestion of such loading and unloading at the main 
freight terminal. The cars are brought to these yards sometimes 
on floats (as is done so extensively at various points around New 
York Harbor), or they are run down on a long siding running 
perhaps through the city streets. But the essential feature of 

» these yards is the maximum utilization of every square foot of 
yard space, which is always very valuable and which is frequently 
of such an inconvenient shape that a great ingenuity is required 
to obtain good results. There is generally a temptation to use 
excessively sharp curves. When the radii are greater than 175 
feet no especial trouble is encountered. Curves with radius as 
short as 50 feet have been used in some yards. On such curves 
the long cars now generally used make a sharper angle with each 
other than that for which the couplers were designed and spe- 
cial coupler-bars become necessary. The two general methods 
of construction are (a) a series of parallel team tracks (as pre- 
viously described and as illustrated further in Fig. 165), and (h) 
the "loop system," as is illustrated in Fig. 166. 

298. Transfer cranes. These are almost an essential feature 
for yards doing a large business. The transportation of built- 
up girders, castings for excessively heavy machinery, etc., which 
weigh five to thirty tons and even more, creates a necessity for 
machinery which w^ill easily transfer the loads from the car to 
the truck andmce versa. An ordinary "gin-pole" will serve the 
purpose for loads which do not much exceed five tons. A fixed 
framework, covering a span long enough for a car track and a 
team space, with a trolley traveling along the upper chord, is the 
next design in the order of cost and convenience. Increasing 
the span so that it covers two car tracks and two team spaces 
will very materially increase the capacity. Making the frame 
movable so that it travels on tracks which are parallel to the 
car tracks, giving the frame a longitudinal motion equal to two 
or three car lengths, and finally operating the raising and travel- 
ing mechanism by power, the facility for rapidly disposing of 



§ 298. 



YARDS AND TERMINALS. 



347 




East 135th St. 

Fig. 166.— Minor Freight Yard on a Harbor Front. 



348 RAILROAD CONSTRUCTION. § 299 

heavy articles of freight is greatly increased. Of course only a 
very small proportion of freight requires such handling, and the 
business of a yard must be large or perhaps of a special character 
to justify and pay for the installation of such a mechanism. 
Figs. 165 and 166 each indicate a transfer crane, evidently of the 
fixed type. 

299. Track scales. The location of these should be on one of 
the receiving tracks near the entrance to the yard, but not on the 
main track. It is always best to have a ''dead track^' over the 
scales — i.e., a track which has one rail on the solid side wall of 
the scale pit and the other supported at short intervals by posts 
which come up through the scale platform and yet do not touch 
it. These rails and the regular scale rails switch into one track 
by means of point rails a few feet beyond each end of the scales. 
The switches should be normally set so that all trains will use 
the dead track, unless the scales are to be operated. It has been 
found possible in a gravity yard to weigh a train with very httle 
loss of time by running each car slowly by gravity over the 
scales and weighing them as they pass over. 

ENGINE YARDS. 

300. General principles. Engine yards must contain all the 
tracks, buildings, structures, and facilities which are necessary 
for the maintenance, care, and storage of locomotives and for pro- 
viding them with all needed supplies. The supplies are fuel, 
water, sand, oil, waste, tallow, etc. Ash-pits are generally neces- 
sary for the prompt and economical disposition of ashes ; engine- 
houses are necessary for the storage of engines and as a place 
where minor repairs can be quickly made. A turn-table is an- 
other all but essential requirement. The arrangement of all 
these facilities in an engine yard should properly depend on the 
form of the yard. In general they should be grouped together 
and should be as near as possible to the place where through en- 
gines drop the trains just brought in and where they couple on 
to assembled outgoing trains, so that all unnecessary running light 
may be avoided. In Figs. 164 and 167 are show^n two designs 
w^hich should be studied with reference to the relative arrange- 
ment of the yard facilities. 



§ 300. 



YARDS AND TERMINALS. 



349 




Pio. 167. — Engine Yabd and Shops, TIrbana. Ili, 



350 RAILROAD CONSTRUCTION. § 300. 



PASSENGER TER:\riNALS. 

(Passenger terminals are one of the logical subdivisions of 
this chapter, but their construction does not concern one engineer 
in a thousand. The local conditions attending their construction 
are so varied that each case is a special problem in itself — a prob- 
lem which demands in many respects the services of the archi- 
tect rather than the engineer. The student who wishes to pursue 
this subject is referred to an admirable chapter iir '^ Buildings and 
Structures of American Railroads/' by Walter G. Berg, Chief 
Engineer of the Lehigh Valley Railroad.) 




(To face p. 350.) (Published Ihrough courtesy of Union Switch and Signal C^ ) 



CHAPTER XIV. 

BLOCK SIGNALING. 
GENERAL PRINCIPLES. 

301. Two fundamental systems. The growth of systems of 
block signaling has been enormous within the last few years — 
both in the amount of it and in the development of greater per- 
fection of detail. The development has been along two general 
lines: (a) the manual, in which every change of signal is the re- 
sult of some definite action on the part of some signalman, but in 
which every action is so controlled or limited or subject to 
the inspection of others that a mistake is nearly, if not quite, 
impossible; (h) the automatic^ in which the signals are oper- 
ated by mechanism, which cannot set a ^\Tong signal as long as the 
mechanism is maintained in proper order. The fundamental 
principles of the two systems will be briefly outlined, after 
which the chief details of the most common systems will be 
pointed out. 

302. Manual systems. Any railroad w^hich has a telegraph 
line and an operator at all regular stations may (and frequently 
does) operate its trains according to the fundamental princi- 
ples of the manual block system even though it makes no claim 
to a block-signal system. The basic idea of such a system is 
that after a train has passed a given telegraph- or signal-station, 
no other train will be permitted to follow it into that "block'' 
until word is telegraphed from the next station ahead that the 
first train has passed out of that block. With a double-track 
road the operation is very simple; trains may be run at short 
intervals ^vith. long blocks; with an average speed of 30 miles 
per hour and blocks 5 miles long, trains could be run on a ten 
minute interval (nearly). A road with any such traffic would, 
of course, have much shorter blocks, and, practically, they 
would need to be considerably shorter. 

With a single-track road the operation is much more complex, 
since the operator must keep himself informed of the move- 

351 



352 RAILROAD CONSTRUCTION. § 302. 

ments of the trains in both directions. The ratio of length of 
block to train interval would be only one half (and practically 
much less than half) what it could be with a double-track road, 
When such a system is adhered to rigidly, it is called an absolute 
block system But when operating on this system, a delay of 
one train will necessarily delay every other train that follows 
closely after. A portion, if not all, of the delay to subsequent 
trains ma}^ be avoided, although at some loss of safety, by a 
system of permissive blocking. By this system an operator 
may give to a succeeding train a "clearance card'' which per- 
mits it to pass into the next block, but at a reduced speed and 
with the train under such control that it may be stopped on 
very short notice, especially near curves. One element of the 
danger of this system is the discretionary power with which it 
invests the signalmen, a discretion which may be wrongfulh' 
exercised. A modification (which is a fruitful source of colli- 
sions on single-track roads) is to order two trains to enter a 
block approaching each other, and with instructions to pass 
each other at a passing siding at which there is no telegraph- 
station. When the instructions are properly made out and 
literally obeyed, there is no trouble, but every thousandth or 
ten thousandth time there is a mistake in the orders, or a mis- 
understanding or disobedience, and a collision is the result. The 
telegraph line, a code of rules, a corps of operators, and sig- 
nals under the immediate control of the operators, are all that 
is absolutely needed for the simple manual system. 

303. Development of the manual system. One great diffi- 
culty with the simple system just described is that each operator 
is practically independent of others except as he may receive 
general or specific orders from a train-dispatcher at the division 
headquarters. Such difficulties are somewhat overcome by a 
very rigid system of rules requiring the signalmen at each station 
to keep the adjacent signalmen or the train-dispatcher in- 
formed of the movements of all trains past their own stations. 
When these rules (which are too extensive for quotation here) 
are strictly observed, there is but little danger of accident, and 
a neglect by any one to observe any rule will generally be appar- 
ent to at least one other man. Nevertheless the safety of trains 
depends on each signalman doing his duty, and a little careless- 
ness or forgetfulness on the part of any one man may cause an 
accident. The signaling between stations may be done by 



J 303. BLOCK SIGNALING. 353 

ordinary telegraphic messages or by telephone, but is frequently 
done by electric bells, according to a code of signals, since these 
may be readil}^ learned by men who would have more difficulty 
in learning the Morse code. 

In order to have the signalmen mutually control each other, 
the "controlled manual" system has been devised. The first 
successful system of this kind which was brought into exten- 
sive use is the "Sykes" system, of which a brief description 
is as follows: Each signal is worked by a lever; the lever is 
locked by a latch, operated by an electro-magnet, which, with 
other necessary apparatus, is inclosed in a box. When a signal 
is set at danger, the latch falls and locks the lever, which cannot 
be again set free until the electro-magnet raises the latch. The 
magnet is energized only by a current, the circuit of which is 
closed by a "plunger" at the next station ahead; just above 
the plunger is an "indicator," also operated by the current, 
which displays the words clear or blocked. (There are varia- 
tions on this detail.) When a train arrives at a block station 
(.4), the signalman should have previously signaled to the station 
ahead (B) for permission to free the signal. The man ahead (B) 
pushes in the "plunger" on his instrument (assuming that the 
previous train has already passed him), which electrically opens 
the lock on the lever at the previous station (A). The signal 
at A can then be set at "safety." As soon as the train has 
passed A the signal at A must be set at " danger." A further 
development is a device by which the mere passage of the train 
over the track for a few^ feet be3^ond the signal will automati- 
cally throw the signal to "danger." After the signal once goes 
to danger, it is automatically locked and cannot be released 
except by the man in advance (B), who will not do so until the 
train has passed him. The "indicator" on ^'s instrument 
shows "blocked" when A^s signal goes to danger after the train 
has passed A, and jB's plunger is then locked, so that he can- 
not release A's signal while a train is in the block. As soon as 
the train has passed A, B should prepare to get his signals ready 
b}^ signaling ahead to C, so that if the block between B and C 
is not obstructed, B may have his signals at "safety" so that 
the train may pass B without pausing. The student should 
note the great advance in safety made by the Sykes system; 
a signal cannot be set free except by the combined action of 
two men, one the man who actually operates the signal and 



354 RAILROAD CONSTRUCTION. § 304. 

the other the man at the station ahead, who frees the signal 
electrically and who by his action certifies that the block im- 
mediately ahead of the train is clear. 

A still further development makes the system still more '' auto- 
matic" (as described later), and causes the signal to fall to dan- 
ger or to be kept locked at danger, if even a single pair of wheels 
comes on the rails of a block, or if a switch leading from a main 
track is opened. 

304. Permissive blocking. "Absolute" blocking renders ac- 
cidents due to collisions almost impossible unless an engineer 
runs by an adverse signal. The signal mechanism is usually 
so designed that, if it gets out of order, it will inevitabl}^ fall to 
"danger," i.e., as described later, the signal-board is counter- 
balanced by a weight which is much heavier. If the wire breaks, 
the countervveight will fall and the board will assume the hori- 
zontal position, which always indicates " danger." But it some- 
times happens that when a train arrives at a signal-station, the 
signalman is unable to set the signal at safety. This may be 
because the previous train has broken down somewhere in the 
next block, or because a s\vitch has been left open, or a rail has 
become broken, or there is a defect of some kind in the electrical 
connections. In such cases, in order to avoid an indefinite 
blocking of the whole traffic of the road, the signalman may 
give the engineer a "caution-card" or a "clearance card," 
which authorizes him to proceed slowly and with his train under 
complete control into the block and through it if possible. If 
he arrives at the next station without meeting any obstruction 
it merely indicates a defective condition of the mechanism, 
which will, of course, be promptly remedied. Usually the next 
section will be found clear, and the train may proceed as usual. 
On roads where the "controlled manual" SA'stem has received 
its highest development, the rules for permissive blocking are 
so rigid that there is but little danger in the practice, unless 
there is an absolute disobedience of orders. 

305. Automatic systems. By the very nature of the case, 
such systems can only be used to indicate to the engineers of 
trains something with reference to the passage of previous 
trains. The complicated shifting of switches and signals which 
is required in the operation of yards and terminals can only be 
accomplished by "manual" methods, and the only automatic 
features of these methods consist in the mechanical checks 



§ 306. BLOCK SIGNALING. 355 

(electric and otherwise), which will prevent wrong combina- 
tions of signals. But for long stretches of the road, where it 
is only required to separate trains by at least one block length, 
an automatic system is generally considered to be more relia- 
ble. As expressed forcibly by a railroad manager, ''an auto- 
matic system does not go to sleep, get drunk, become insane, 
or tell lies when there is any trouble." The same cannot always 
be said of the employes of the manual system. 

The basic idea of all such systems is that when a train passes 
a signal-station (A), the signal automatically assumes the ''dan- 
ger" position. This may be accomplished electrically, pneu- 
matically, or even by a direct mechanism. When the train 
reaches the end of the block at B and passes into the next one, 
the signal at B will be set at danger and the signal at A will be 
set at safety. The lengths of the blocks are usually so great 
that the only practicable method of controlling from B a 
mechanism at A is by electricity, although the actual motive 
power at A may be pneumatic or mechanical. At one time 
the current from A to ^ was carried on ordinary wires. This 
method has the very positive advantage of reliability, definite 
resistance to the current, and small probability of short-circuit- 
ing or other derangement. But now all such systems use the 
rails for a track circuit and this makes it possible to detect the 
presence of a single pair of wheels on the track anywhere in the 
block, or an open switch, or a broken rail. Any such circum- 
stances, as well as a defect in the mechanism, will break or 
short-circuit the current and will cause the signal to be set at 
danger. To prevent an indefinite blocking of traffic owing to 
a signal persistently indicating danger, most roads employing 
such a system have a rule substantially as follows : When a train 
finds a signal at danger, after waiting one minute (or more, 
depending on the rules), it may proceed slowly, expecting to 
find an obstruction of some sort; if it reaches the next block 
without finding any obstruction and finds the next signal clear, 
it may proceed as usual, but must promptly report the case to 
the superintendent. Further details regarding these methods 
will be given later. See § 310. 

306. "Distant" signals. The close running of trains that 
is required on heavy-traffic roads, especially where several 
branches combine to enter a common terminal, necessitates the 
use of very short blocks. A heavy train running at high speed 



356 RAILROAD CONSTRUCTION. § 306. 

can hardly make a "service'' stop in less than 2000 feet, while 
the curves of a road (or other obstructions) frequentl}^ make 
it difficult to locate a signal so that it can be seen more than a 
few hundred feet away. It would therefore be impracticable 
to maintain the speed now used with heavy trains if the engi- 
neer had no foreknowledge of the condition in which he will 
find a signal until he arrives wdthin a short distance of it. To 
overcome this difficulty the "distant" signal was de^dsed. This 
is placed about 1800 or 2000 feet from the "home" signal, and 
is interlocked with it so that it gives the same signal. The dis- 
tant signal is frequently placed on the same pole as the home 
signal of the previous block. When the engineer finds the 
distant signal "clear," it indicates that the succeeding home 
signal is also clear, and that he may proceed at full speed and 
not expect to be stopped at the next signal; for the distant 
signal cannot be cleared until the succeeding home signal is 
cleared, which cannot be done until the block succeeding that 
is clear. A clear distant signal therefore indicates a clear track 
for two succeeding blocks. When the engineer finds the distant 
signal blocked, he need not stop (providing the home signal is 
clear). It simply indicates that he must be prepared to stop 
at the next home signal and must reduce speed if necessary. 
It may happen that by the time he reaches the succeeding home 
signal it has already been cleared, and he may proceed without 
stopping. This device facilitates the rapid running of trains, 
with no loss of safety, and yet with but a moderate addition to 
the signaling plant. 

307. "Advance '' signals. It sometimes becomes necessary 
to locate a signal a few hundred feet short of a regular passen- 
ger-station. A train might be halted at such a signal because 
it was not cleared from the signal-station ahead — perhaps a 
mile or two ahead. For convenience, an "advance" signal 
may be erected immediately beyond the passenger-station. 
The train will then be permitted to enter the block as far as 
the advance signal and may deliver its passengers at the station. 
The advance signal is interlocked with the home signal back 
of it, and cannot be cleared until the home signal is cleared and 
the entire block ahead is clear. In one sense it adds another 
block, but the signal is entirely controlled from the signal station 
back of it. 



p 



§ 308. BLOCK SIGNALING. 357 



MECHANICAL DETAILS. 

308. Signals. The primitive signal is a mere cloth flag. A 
better signal is obtained when the flag is suspended in a suit- 
able place from a fixed horizontal support, the flag weighted 
at the bottom, and so arranged that it may be drawn up and 
out of sight by a cord which is run back to the operator's office. 
The next step is the substitution of painted wood or sheet metal 
for the cloth flag, and from this it is but a step to the standard 
semaphore on a pole, as is illustrated in Fig. 168. The simple 
flag, operated for convenience with a cord, is the signal em- 
ployed on thousands of miles of road, where they perhaps make 
no claim to a block-signal system, and yet w^here the trains 
are run according to the fundamental rules of the simple manual 
block method. 

Semaphore boards. These are about 5 feet long, 8 inches 
wide at one end, and tapered to about 6 inches wide at the hinge 
end. The boards are fastened to a casting which has a ring to 
hold a red glass which may be swung over the face of a lantern, 
so as to indicate a red signal. ^'Distant" signal-boards usually 
have their ends notched or pointed; the ^'home'' signal-boards 
are square ended. The boards are always to the right of the 
hinge when a train is approaching them. The ''home" signals 
are generally painted red and the '^ distant" signals green, 
although these colors are not invariable. The backs of the 
boards are painted white. Therefore any signal-board which 
appears on the left side of its hinge will also appear white, and 
is a signal for traffic in the opposite direction, and is therefore 
of no concern to an engineman. 

Poles and bridges. When the signals are set on poles, they 
are generall}^ placed on the right-hand side of the track. When 
there are several tracks, four or more, a bridge is frequently 
built and then each signal is over its own track. When switches 
run off from a main track, there may be several signal-boards 
over one track. The upper one is the signal for the main track 
and the lower ones for the several switches. In Fig. 169 is 
shown a ''bridge" with its various signal-boards controlling the 
several tracks and the switches running off from them. 

"Banjo" signals. This name is given to a form of signal, 
illustrated in Fig. 170, in which the indication is taken from the 



358 RAILROAD CONSTRUCTION. § 308. 

color of a round disk inclosed with glass. This is the distinctive 
signal of the Hall Signal Company, and is also used by the 
Union Switch and Signal Company. The great argument in 
their favor is that they may be worked b}^ an electric current 
of low voltage, which is therefore easily controlled; that the 
mechanism is entirely inside of a case, is therefore very light, 
and is not exposed to the weather. The argument urged 
against them is that it is a signal of color rather than form 
or 'position, and that in foggy A\eather the signal cannot be 
seen so easily; also that unsuspected color-blindness on the 
part of the engineman may lead to an accident. Notwith- 
standing these objections, this form of signal is used on thousands 
of miles of line in this country. 

309. Wires and pipes. Signals are usually operated by levers 
in a signal-cabin, the levers being very similar to the reversing- 
lever of a locomotive. The distance from the levers to the sig- 
nals is, of course, very variable, but it is sometimes 2000 feet. 
The connecting-link for the most distant signals is usually 
No. 9 wire; for nearer signals and for all switches operated 
from the cabin it may be 1-inch pipe. When not too long, one 
pipe will serve for both motions, forward and back. When 
wires are used, it is sometimes so designed (in the cheaper sys- 
tems) that one wire serves for one motion, gravity being de- 
pended on for the other, but now all good systems require two 
wares for each signal. 

Compensators. Variations of temperature of a material with 
as high a coefficient as iron will cause very appreciable differ- 
ence of length in a distance of several hundred feet, and a. 
dangerous lack of adjustment is the result. To illustrate: A 
fall of 60° F. will change the length of 1000 feet of wire by 

1000 X 60 X .0000065 = 0.39 foot = 4.68 inches. 

A much less change than this will necessitate a readjustment 
of length, unless automatic compensators are used. A com- 
pensator for pipes is very readil}^ made on the principle illus- 
trated in Fig. 171. The problem is to preserve the distance 
between a and d constant regardless of the temperature. Place 
the compensator half-way between a and d, or so that oh = cd. 
A fall of temperature contracts ah to ah' . Moving h io ¥ will 
cause c to move to c', in which hb' = cc\ But cd has also short- 
ened to c'd; therefore d remains fixed in position. 



(To face page 358.) 




Courtesy of the U)U( 



Sicitrh niiif Signal Co. 

Fig. 168. — Semaphores, 




C3 







{To face page 358.) 




-f^ 





^^F*^ 



r1 



A 



Cuix, te^y u-j ,'/ic L iiiun switch and Signal Co, 

Fig. 170. — " Banjo " Signals. 



§309. 



BLOCK SIGNALING. 



359 



The regulations of the A. R. E. & M. W. Assoc, require that 
"A compensator shall be provided for each pipe line over fifty 
(50) feet in length and under eight hundred (800) feet, with 
crank-arms eleven by thirteen (11X13) inch centers. From 
eight hundred (800) to twelve hundred (1200) feet in length, 
crank-arms shall be eleven by sixteen (11X16) inch centers. 
Pipe lines over twelve hundred (1200) feet in length shall be 
provided with an additional compensator. 

''Compensators shall have one sixty (60) degree and one one 
hundred and twenty (120) degree angle-cranks and connecting 



a h b 




Fi(5. 171. — Standard Pipe Compensator. 



link, mounted in cast iron base, having top of center pins sup- 
ported. The distance between center of pin-holes shall be 
twenty-two (22) inches. '' 

The compensator should be placed in the middle of the length 
when only one is used. When two are used they should be 
placed at the quarter points. Note that in operating through 
a compensator the direction of motion changes; i.e., if a moves 
to the right, d moves to the left, or if there is compression in ah 
there is tension in cd, and vice versa. Therefore this form of 
compensator can only be used with pipes which will withstand 
compression. It has seemed impracticable to design an equally 



360 



RAILROAD CONSTRUCTION. 



309. 



satisfactory compensator for wires, although there are several 
designs on the market. 

The change of length of these bars is so great that allowance 
must be made for the temperature at the time of installation. 
On the basis of 50° as the mean temperature, the pipes are so 
adjusted that the distance between the points b and c of Fig. 171 
is made greater or less than 22 inches, according to the tem- 
perature of installation. For example, if the temperature were 
80° and the length of the piping were 900 feet, the length of the 
pipes should be adjusted so that be is less than 22 inches by an 
amount equal to 900 X (80°- 50°) X .0000065 = 0.1755 feet = 
2.106 inches. The length should therefore be 19.9 inches in- 
stead of 22 inches. If the mean temperature was very different 
(say in Florida) some higher temperature should be taken as 
normal, so that the extreme range above and below the normal 
shall be approximately the same. 

Guides around curves and angles. When wires are required 
to pass around curves of large angle, pulleys are used, and a 
length of chain is substituted for the wire. For pipes, when 
the curve is easy the pipes are slightly bent and are guided 
through pulleys. When the angle is sharper, "angles" are 
used. The operation of these details is self-evident from an 
inspection of Fig. 172. 

310. Track circuit for automatic signaling. The several 
systems of automatic signaling differ in the minor details, but 
nearly all of them agree in the following particulars. A current 
of low potential is run from a battery at one end of a section 
through one line of rails to the other end of the section, then 
through a relay, and then back to the battery through the other 




Fig. 172. — Deflecting-rods. 



line of rails. To avoid the excessive resistance which would 
occur at rail joints which may become badly rusted, a wire 



§ 310. 



BLOCK SIGNALING. 



361 



suitably attached to the rails is run 
order to insulate the rails of one sec- 
tion from the rails at either end and 
yet maintain the rails structurally con- 
tinuous, the ends of the rails at these 
dividing points are separated by an 
insulator and the joint pieces are either 
made of wood or have som3 insulating 
material placed between the rails and 
the ordinary metal joint. The bolts 
must also be insulated. When the 
relay is energized by a current, it 
closes a local circuit at the signal- 
station, which will set the signal there 
at "safety." Ths resistance of the 
relay is such that it requires nearly the 
whole current to work it and to keep 
the local circuit closed. Therefore, 
when there is any considerable loss of 
current from one rail to the other, the 
relay will not be sufficiently energized, 
the local circuit will be broken, and the 
signal will automatically fall to danger. 
This diversion of current from one rail 
to the other before the current reaches 
the relay may be caused in several 
ways: the presence of a pair of wheels 
on the rails Anywhere in the section will 
do it ; also the breakage of a rail ; also 
the opening of a switch anywhere in 
the section ; also the presence of a pair 
of wheels on a siding between the 
"fouling point'' and the switch. (The 
"fouling point" of a siding is that 
point where the rails first commence to 
approach the main track.) In Fig. 173 
is shown all of the above details as well 
as some others. At A, J5, and the 
"fouling point" are shown the in- 
sulated joints. The batteries and 
signals are arranged for train motion 



around each joint. In 




iWli 




Fia. 173. 



362 RAILROAD CONSTRUCTION. § 310. 

to the right. When a train has passed the points near A, where 
the wires leave the rails for the relay, the current from the 
"track battery'' at B will pass through the wheels and axles, and 
although no electrical connection is broken, so much current 
will be shunted through the wheels and axles that the weak 
current still passing through the relay is not strong enough to 
energize it against its spring and the "signal-magnet" circuit 
is broken, and the signal A goes to "danger." At the turnout 
) the rails between the fouling point and the switch are so con- 
nected (and insulated) that a pair of wheels on these rails will 
produce the same effect as a pair on the main track. This is 
to guard against the effect of a car standing too near the switch, 
even though it is not on the main track. When the train passes 
Bj if there is no other interruption of the current, the track 
battery at B again energizes the relay at A, the signal-magnet 
circuit at A is closed, and the signal is draw^n to "safety." 

(The present edition has omitted several subdivisions of this 
general subject, notably the "staff system," used chiefly in 
England, and all discussions of "interlocking'' which is an 
essential feature of the operation of large terminal yards. A 
future edition may supply these deficiencies, although an ex- 
haustive treatment of the subject of Signaling would require a 
separate volume.) 



CHAPTER XV. 



ROLLING-STOCK. 



(It is perhaps needless to say that the following chapter is 
in no sense a course in the design of locomotives and cars. Its 
chief idea is to give the student the elements of the construc- 
tion of those vehicles which are to use the track which he may 
design — to point out the mutual actions and reactions of vehicle 
against track and to show the effect on track wear of varia- 
tions in the design of rolling-stock. The most of the matter 
given has a direct practical bearing on track-work, and it is con- 
sidered that all of it is so closely related to his work that the 
civil engineer may study it with profit.) 



WHEELS AND RAILS. 

311. Effect of rigidly attaching wheels to their axles. The 

wheels of railroad rolling-stock are invariably secured rigidly 
to the axles, which therefore revolve with the wheels. The 
chief reason for this is to avoid excessive wear 
between the axles and the wheels. 

Any axle must always be somewhat loose in 
its journals. A side wise force P (see Fig. 174) 
acting against the circumference of the wheel 
will produce a much greater pressure on the 
axle at S and S' , and if the wheel moves on 
the axle, the wear at S and S^ will be exces- 
sive. But w^hen the axle is fitted to the wheel 
with a ^^ forced fif and does not revolve, 
the mere pressure produced at >S is harmless. 
When two wheels are fitted tight to an axle, 
as in Fig. 175, and the axle revolves in the jour- Fiq. 174. 

nals aa, a side wise pressure of the rail against the wheel flange 
will only produce a slight and harmless increase of the journal 
pressure Q, although at Q there is sliding contact. Twist- 

363 




364 



BAILROAD CONSTRUCTION. 



§ 312. 



ing action in the journals is thus practically avoided, since a 
small pressure at the journal-boxes at each end of the axle 
suffices to keep the axle truly in line. 



a 1= 
"P 



Fig. 175. 







Fig. 176. 



On the other hand, when the wheels are rigidly attached to 
their axles, both wheels must turn together, and when rounding 
curves, the inner rail being shorter than the outer rail, one 
wheel must slip by an amount equal to that difference of length. 
The amount of this slip is readily computable : 



Longitudinal slip = ^^{r^ - ^i) = 



360° 



' = €0!" 



(136) 



in which C is a constant for any one gauge, and g= the track 
gauge = (r2 — ri). For standard gauge (4.708) the shp is .08218 
foot per degree of central angle. This shows that the longitu- 
dinal slipping around any curve of any given central angle will 
be independent of the degree of the curve. The constant (.08218) 
here given is really somewhat too small, since the true gauge 
that should be considered is the distance between the lines of 
tread on the rails. This distance is a somewhat indeterminate 
and variable quantity, and probably averages 4.90 feet, which 
would increase the constant to .086. The slipping may occur 
by the inner wheel slipping ahead or the outer wheel slipping 
back, or by both wheels slipping. The total slipping will be 
constant in any case. The slipping not only consumes power, 
but wears both the w^heels and the rail. But even these dis- 
advantages are not sufficient to offset the advantages resulting 
from rigid w^heels and axles. 

312. Effect of parallel axles. Trucks are made with two or 
three parallel axles (except as noted later), in order that the 
axles shall mutually guide each other and be kept approximately 



§312. 



ROLLING-STOCK. 



365 



perpendicular to the rails. If the curvature is very sharp and 
the wheel-base comparatively long (as is notably the case on 
street railways at street corners), the front and rear wheels 





Fig. 177. 



Fig. 178. 



d.'-' 

^^^ 



will stand at the same angle (a) with the track, as shown in 
Fig. 177. But it has been noticed that for ordinar\^ degrees of 
curvature, the rear wheels stand radial to the curve (see Fig. 
178), and for steam railroad work this is the normal case. When 
the t^\'0 parallel axles are on a curve (as shown) ^ the wheels tend 
to run in a straight line. In order that they shall run on a curve 
they must slip laterally. The principle 
is illustrated in an exaggerated form in 
Fig. 179. The wheel tends to roll from a ^ 
toward b. Therefore in passing along the 
track from a to c it must actually slip late- '' 
rally an amount he which equals ac sin a. ^^^' ^''^• 

Let / = length of the wheel-base (Figs. 177 and 178); r = radius 
of curve; then for the first case (Fig. 177), sina = ^-^2r; for 
the second and usual case (Fig. 178), sin a = t-^r; for t = 5 feet 
and r = radius of a 1° curve, a = 0°03' for the second case, a 
varies (practically) as the degree of curve. The lateral slipping 
per unit of distance traveled therefore equals sin a. As an 
illustration, given a 5-foot wheel-base on a 5° curve, a = 0° 15', 
sin a = .00436, and for each 100 feet traveled along the curve 
the lateral slip of the front wheels would be 0.436 foot. There 
would be no lateral slipping of the rear wheels, assuming that 
the rear axle maintained itself radial. 

From the above it might be inferred that the flanges of the 
forward wheels will have much greater wear than those of the 
rear wheels. Since cars are drawn in both directions about 
equally, no difference in flange wear due to this cause will occur, 
but locomotives (except switching-engines) run forward almost 



366 



RAILROAD COXSTRXJCTION. 



§313. 



exclusively, and the excess wear of the front wheels of the pilot - 
and tender-trucks is plainly observable. 

For a given curve the angle a (and the accompanying resist- 
ance) is evidently greater the greater the distance between 
the axles. On the other hand, if the two axles are very close 
together, there will be a tendency for the truck to twist and 
the wheels to become jammed, especially if there is consider- 
al)le play in the gauge. The flange friction would be greater 
and would perhaps exceed the saving in lateral slipping. A 
general rule is that the axles should never be closer together 
than the gauge. 

Although the slipping per unit of length along the curve varies 
directly as the degree of curvature, the length of curve necessary 
to pass between two tangents is inversely as the degree of curve, 
and the total slipping between the two tangents is independent 
of the degree of curve. Therefore when a train passes between 

two tangents, the total slipping 
of the wheels on the rails, lon- 
gitudinal and lateral, is a quantity 
which depends only on the central 
il LJ angle and is independent of the 

_ JL__ _^-L... radius or degree of curve. 

313. Effect of coning wheels. 

The wheels are alwaj^s set on the 

axle so that there is some '^play" 

or chance for lateral motion be- 

|i j I tween the wheel-flanges and the 

^pJ [ U--^ rail. The treads of the wheel are 

. ^ .. -j^ ^^^^ ^^ coned." This coning and play 

of gauge are sho^^'n in an exagger- 
ated form in Fig. 180. When the 
wheels are on a tangent, although there will be occasional oscil- 
lations from side to side, the normal position will be the sym- 
metrical position in which the circles of tread hh are equal. 
When centrifugal force throvrs the wheel-flange against the rail, 
the circle of tread a is larger than 6, and much larger than c; 
therefore the wheels will tend to roll in a circle whose radius 
equals the slant height of a cone whose elements would pass 
through the unequal circles a and c. If this radius equaled the 
radius of the track, and if the axle w^ere free to assume a radial 
position, the wheels would roll freely on the rails without an}^ 



rt=t 






Fig. 180. 



§ 314. ROLLING-STOCK. 367 

slipping or flangie pressure. Under such ideal conditions, 
coning would be a valuable device, but it is impracticable to 
have all axles radial, and the radius of curvature of the track 
is an extremely variable quantity. It has been demonstrated 
that with parallel axles the influence of coning diminishes as 
the distance between the axle increases, and that the effect is 
practically inappreciable when the axles are spaced as they are 
on locomotives and car-trucks. The coning actuall}^ used is 
very slight (see Chapter XV, § 332) and has a different object. 
It is so slight that even if the axles were radial it would only 
prevent the slipping on a very light curve — say a 1° curve. 

314. Effect of flanging locomotive driving-wheels. If all the 
wheels of all locomotives were flanged it would be practically 
impossible to run some of the longer types around sharp curves. 
The track-gauge is always widened on curves, and especially 
on sharp curves, but the widening would need to be excessive 
to permit a consolidation locomotive to pass around an 8° or 
10° curve if all the drivers were flanged. The action of the 
wheels on a curve is illustrated in Figs. 181, 182, and 184. All 
small truck-wheels are flanged. The rear driA^ers are always 
flanged and four-driver engines usually have all the drivers 
flanged. Consolidation engines have only the front and rear 
drivers flanged. Mogul and ten-wheel engines have one pair 
of drivers blank. On Mogul engines it is always the middle 
pair. On ten-wheel engines, when used on a road having sharp 
curves, it is preferable to flange the front and rear driving- 
wheels and use a ^^ swing bolster" (see § 315); when the curva- 
ture is easy, the middle and rear drivers may be flanged and 
the truck made with a rigid center. The blank drivers have 
the same total width as the other drivers and of course a much 
wider tread, which enables these drivers to remain on the rail, 
even though the curvature is so sharp that the tread overhangs 
the rail considerably. 

315. Action of a locomotive pilot- truck. The purpose of 
the pilot-truck is to guide the front end of a locomotive around 
a curve and to relieve the otherwise excessive flange pressure 
that would be exerted against the driver-flanges. There are 
two classes of pilot-trucks — (a) those having fixed centers and 
(b) those having shifting centers. This second class is again 
subdivided into two classes, which are radically different in 
their action — (b^) four-wheeled trucks having two parallel axles 



368 



RAILROAD CONSTRUCTION. 



§315. 



and (62) two-wheeled trucks which are guided by a ''radius- 
bar." The action of the four-w^heeled fixed-centered truck (a) 
is shown in Fig. 181. Since the center of the truck is forced 




Fig. 181. — Fixed Center Pilot-truck. 
to be in the center of the track, the front drivers are drawn 
away from the outer rail. The rear outer driver tends to roll 
away from the outer rail rather than toward it, and so the effect 




Fig. 182. — Four-wheeled Truck — Shifting Center. 

of the truck is to relieve the driver-flanges of any excessive 
pressure due to curvature. The only exception to this is the 
case where the curvature is sharp. Then the front inner driver 
may be pressed against the inner rail, as indicated in Fig. 181. 

This limits the use of this type of 
wheel-base on the sharper curves. 
The next type — (h^) four-wheeled 
trucks with shifting centers — is 
much more flexible on sharp 
curvature; it likewise draws the 
front drivers away from the outer 
rail. The relative position of the 
wheels is sho\ATi in Fig. 182, in 
which c' represents the position 
of center-pin and c the displaced 
truck center. The structure and 
action of the truck is shown in 
Fig. 183. The "center-pin" (1) is 
supported on the ''truck-bolster" (2), which is hung by the 
"links" (4) from the "cross-ties" (3). The Jinks are therefore 




Fig, 



183. — Action of Shifting 
Center. 



§315. 



ROLLING-STOCK. 



3G9 



in tension and when the wheels are forced to one side by the 
rails the links are inclined and the front of the engine is 
drawn inw^ard by a force equal to the weight on the bolster 
times the tangent of the angle of inclination of the links. This 
assumes that all links are vertical when the truck is in the 
center. Frequently the opposite links are normally inclined to 
each other, which somewhat complicates the above simple relation 
of the forces, although the general principle remains identical. 

The two-wheeled pilot-truck with shifting center is illus- 
trated in Fig. 184. The figure shows the facility with which 





Fig. 184. — Two-wheeled Truck — Shifting Center. 
an engine with long wheel-base may be made to pass around 
a comparatively sharp curve by omitting the flanges from the 
middle drivers and using this form of pilot-truck. As in the 
previous case, the eccentricity of 
the center of the truck relative 
to the center-pin induces a cen- 
tripetal force which draws the 
front of the engine inward. But 
the swing- truck is not the only 
source of such a force. If the 
''radius-bar pin" were placed at 0' (see Fig. 185), the truck- 
axle would be radial. But the radius-bar is always made some- 
what shorter than this, and the pin is placed at 0, a considerable 
distance ahead of O', thus creating a tendency for the truck 
to run toward the inner rail and draw the front of the loco- 
motive in that direction. This tendency will be objectionably 
great if the radius-bar is made too short, as has been practically 
demonstrated in cases when the radius-bar has been subse- 
quently lengthened with a resulting improvement in the running 
of the engine. 



Fig. 185. — Action of Two- 
wHEf^ED Truck. 



370 



RAILROAD COXSTRUCTIOX. 



§ 316. 



LOCOMOTIVES. 
GENERAL STRUCTURE. 

316. Frame. The frame or skeleton of a locomotive con- 
sists chiefly of a collection of forged wrought-iron bars, as 
shown in Figs. 186 and 187. These bars are connected at the 



QC 



^F 



Fig. 186. — Engine -frame. 

front end by the ''bumper^' (c), which is usually made of wood. 
A little further back they are rigidly connected at bb by the 
cjdinders and boiler-saddle. The boilers rest on the frames 
at aaaa by means of ^'pads," which are bolted to the fire-box, 
but which permit a free expansion of the boiler along the frame. 
This expansion is sometimes as much as //'• ^^ ^ ^'con- 
solidation'' engine (frame shown in Fig. 187) it is frequently 






Fig. 187. — Engine -frame — Consolidation Type. 

necessary to use vertical swing-levers about 12'' long instead 
of "pads." The swinging of the levers permit all necessary 
expansion. At the back the frames are rigidly connected by 
the iron '^ foot-plate." The driving-axles pass through the 
"jaws" dddd^ which hold the axle-boxes. The frame-bars, 
have a width (in plan) of 3'' to 4". The depth (at a) is about 
the same. Fig. 186 shows a frame for an "American" type 
of locomotive; Fig. 187 shows a frame for a ' ^ Consolidation " 
type (see § 323). 

317. Boiler. A boiler is a mechanism for transferring the 
latent heat of fuel to water, so that the water is transformed 
from cold water into high-pressure steam, which by its expan- 
sion will perform work. The efficiency of the boiler depends 
largely on its ability to do its work rapidly and to reduce to 
a minimum the waste of heat through radiation. The boiler 
contains a fire-box (see Fig. 188), in which the fuel is burned. 
The gases of consumption pass from the fire-box through the 
numerous boiler-tubes into the "smoke-box" S and out through 
the smoke-stack. The fire-box consists of an inner and outer 



§ 317. 



ROLLIXG-STOCK. 



371 



shell separated by a layer of water about 3" thick. The ex- 
posure of water-surface to the influence of the fire is thus very 
complete. The efRciency of this transferal of heat is somewhat 
indicated by the fact that, although the temperature of the 
gases in the fire-box is probably from 3000° to 4000° F., the 
temperature in the smoke-box is generally reduced to 500° to 




Fig. 188. — Locomotive-boiler. 
600° F. If the steam pressure is 180 lbs., the temperature of 
the water is about 380° F., and, considering that heat will not 
pass from the gas to the water unless the gas is hotter than the 
water, the water evidently absorbs a large part of the theo- 
retical maximum. Nevertheless gases at a temperature of 
600° F. pass out of the smoke-stack and such heat is utterly 
wasted. 

The tubes vary from If" to 2" , inside diameter, with a thick- 
ness of about O'MO to 0'M2. The aggregate cross-sectional 
area of the tubes should be about one eighth of the grate area. 
The number will vary from 140 to 250. They are made as long 
as possible, but the length is virtually determined by the type 
and length of engine. 

318. Fire-box. The fire-box is surrounded by water on the 
four sides and the top, but since the water is subjected to the 
boiler pressure, the plates, which are about -f^" thick, must be 
stayed to prevent the fire-box from collapsing. This is easily 
accomplished over the larger part of the fire-box surface by 
having the outside boiler-plates parallel to the fire-box plates 
and separated from them by a space of about 3". The plates 
are then mutually held by ''stay-bolts." See Fig. 189. These 
are about y in diameter and spaced 4" to 4 J". The f^" hole, 



372 



RAILROAD COXSTRUCTION. 



§318. 



drilled Ij" deep, indicated in the figure, will allow the escape 
of steam if the bolt breaks just behind the plate, and thus calls 
attention to the break. The stay-bolts are turned down to a 
diameter equal to that at the root of the screw-threads. This 
method of supporting the fire-box sheets is used for the two 
sides, the entire rear, and for the front of the fire-box up to the 
boiler-barrel. The ''furnace tube-sheet" — the upper part of 
the front of the fire-box — is stayed by the tubes. But the top 
of the fire-box is troublesome. It must always be covered 
with water so that it will not be "burned" by the intense heat. 
It must therefore be nearly, if not quite, flat. There are three 
general methods of accomplishing this. 




Fig. 189. 



Fig. 190. 



(a) Radial stays. This construction is indicated in Fig. 190. 
Incidentally there is also shown the diagonal braces for resist- 
ing the pressure on the back end of the boiler above the fire- 
box. It may be seen that the stays are not perpendicular to 
either the crown-sheet or the boiler-plate. This is objection- 
able and is obviated by the other methods. 

(b) Crown-bars. These bars are in pairs, rest on the side 
furnace-plates, and are further supported by stays. See Fig. 
191. 

(c) Belpaire fire-box. The boiler above the fire-box is rect- 
angular, with rounded corners. The stays therefore are per- 
pendicular to the plates. See Fig. 192. 

Fire-brick arches. These are used, as sho\vn in Fig. 193, to 
force all the gases to circulate through the upper part of the fire- 



§ 318. 



ROLLING-STOCK. 



37: 



box. Perfect combustion requires that all the carbon shall be 
turned into carbon dioxide, and this is facilitated by the 
forced circulation. > 




Water-tables. The same object is attained by using a water- 
table instead of a brick arch— as shown in Fig. 191. But it has 



374 



RAILROAD COXSTRUCTION. 



§ 319. 



the iurther advantages of giving additional heating-surface and 
avoiding the continual expense of maintaining the bricks. One 




Fig. 192. — "Belpaire" Fire-box. 
Half -section through AB. Half -section through CD. 



feature of the design is the use of a number of steam-jets 
which force air into the fire-box and assist the combustion. 





Fig. 1 93. — Fire-brick Arch. 



Fig. 194. — Wootten Fire-box. 



Area. Fire-boxes are usually limited in width to the prac_ 
ticable width between the wheels — thus giving a net inside 
width of about 3 feet and a maximum length of 10 to 11 feet — 
this being about the maximum distance over which the firemen 
can properly control the fire. About 37 square feet is the 
maximum area obtainable except when the ''Wootten'' fire- 
box is used — illustrated in Fig. 194. Here the grate is raised 
above the dri^4ng-wheels and has (in the case shown) a width 
of 8' OJ". The fire-box area is over 76 square feet. Note that 
two furnace-doors are used. 

319. Coal consumption. No form of steam-boiler (except 
a boiler for a steam fire-engine) requires as rapid production 
of steam, considering the size of the boiler and fire-box, as a 



§ 319. ROLLING-STOCK. 375 

locomotive. The combustion of coal per square foot of grate 
per hour for stationary boilers averages about 15 to 25 lbs. and 
seldom exceeds that amount. An ordinary maximum for a 
locomotive is 125 lbs. of coal per square foot of grate-area per 
hour, and in some recent practice 220 lbs, have been used. Of 
course such excessive amounts are wasteful of coal, because 
a considerable percentage of the coal will be blown out of the 
smoke-stack unconsumed, the draft necessary for such rapid 
consumption being very great. The only justification of such 
rapid and wasteful coal consumption is the necessity for rapid 
production of steam. The best quality of coal is capable of 
evaporating about 14 lbs. of water per pound of coal, i.e., change 
it from water at 212° to steam at 212°; the heat required to 
change water at ordinary temperatures to steam at ordinary 
working pressure is (roughly) about 20% more. From 6 to 9 lbs. 
of water per pound of coal is the average performance of ordinary 
locomotives, the efficiency being less with the higher rates of 
combustion. Some careful tests of locomotive coal consump- 
tion gave the following figures: when the consumption of coal 
was 50 lbs. per square foot of grate-area per hour, the rate of 
evaporation was 8 lbs. of water per pound of coal. When the 
rate of coal consumption was raised to 180, the evaporation 
dropped to 5^ lbs. of water per pound of coal. It has been 
demonstrated that the efficiency of the boiler is largely increased 
by an increased length of boiler-tubes. The actual consump- 
tion of coal per mile is of course an exceedingly variable quan- 
tity, depending on the size and type of the engine and also on 
the work it is doing — whether climbing a heavy grade with its 
maximum train-load or running easily over a level or down 
grade. A test of a 50-ton engine, running without any train at 
about 20 to 25 miles per hour, showed an average consumption 
of 21 lbs. of coal per mile. Statistics of the Pennsylvania Rail 
road show a large increase (as might be expected, considering 
the growth in size of engines and weight of trains) in the aver- 
age number of pounds of coal burned per ^ram-mile — some of 
the figures being 55 lbs. in 1863, 72 lbs. in 1872, and nearly 
84 lbs. in 1883. Figures are published showing an average 
consumption of about 10 lbs. of coal per passenger-car mile, 
and 4 to 5 lbs. per freight-car mile. But these figures are always 
obtained by dividing the total consumption per train-mile by 
the number of cars, the coal due to the weight of the engine 



376 RAILROAD CONSTRUCTION. § 320, 

being thrown in Wellington developed a rule, based on the 
actual performance of a very large number of passenger-trains, 
that the number of pounds of coal per mile =21.1 4-6.74 times 
the number of passenger-cars The amount of coal assigned 
to the engine agrees remarkably with the test noted above 
For freight-trains the amount assigned to the engine should 
be much greater (since the engine is much heavier), and that 
assigned to the individual cars much less, although the great 
increase in freight-car weights in recent years has caused an 
increase in the coal required per car, 

320. Heating- surface. The rapid production of steam re- 
quires that the hot gases shall have a large heating-surface to 
which they can impart their heat, From 50 to 75 square feet 
of heating-surface is usually designed for each square foot of 
grate-area. A more recently used rule is that there should be 
from 60 to 70 square feet of tube heating-surface per square 
foot of grate-area for bituminous coal. 40 or 50 to 1 is more 
desirable for anthracite coal Almost the w^hole surface of 
the fire-box has w^ater behind it, and hence constitutes heating- 
surface. Although this surface forms but a small part of the 
total (nominally), it is really the most effective portion, since 
the difference of temperature of the gases of combustion and 
the water is here a maximum, and the fiow^ of heat is therefore 
the most rapid. The heating-surface of the tubes varies from 
85 to 93% of the total, or about 7 to 15 times the heating-sur- 
face in the fire-box. Sometimes the heating-surface is as much 
as 2300 square feet, but usually it is less than 2000, even for 
engines which must produce steam rapidly. 

Some of the most recent locomotives have greatly exceeded 
these figures One just constructed for the New York Central 
and Hudson Rivei Railroad has the following figures* heating- 
surface, 3500 sq. ft ; grate-area, 50 sq ft ; cylinders, 21" X 26"; 
total weight, 176000 lbs : weight on drivers, 95000 lbs.; drivers, 
79" diameter; with 85% of the boiler pressure, it developed 
an adhesion of 24700 lbs., which represented a factor of adhesion 

Another rule used by designers is that the engine should 
have 1 sq ft of heating-surface for each 50 or 60 lbs of weight, 
efficiency being mdicated by a low weight- For the above 
engine the ratio is 53. 



§ 321. ROLLING-STOCK. 377 

321. Loss of efficiency in steam pressure. The effective 
work done by the piston is never equal to the theoretical energy 
contained in the steam withdrawn from the boiler. This is due 
chiefly to the following caiises: 

(a) The steam is ''wire-drawn," i.e., the pressure in the 
cylinder is seldom more than 85 to 90% of the boiler pressure. 
This is due largely to the fact that the steam-ports are so small 
that the steam cannot get into the cylinder fast enough to exert 
its full pressure. It is often purposely wire-drawn by partially 
closing the throttle, so that the steam may be used less rapidly. 

(5) Entrained water. Steam is always drawn from a dome 
placed over the boiler so that the steam shall be as far above 
the water-surface as possible, and shall be as dry as possible. 
In spite of this the steam is not perfectly dry and carries with 
it water at a temperature of, say, 361°, and pressure of 140 lbs. 
per square inch. When the pressure falls during the expan- 
sion and exhaust, this hot water turns into steam and absorbs 
the necessary heat from the hot cylinder-walls. This heat is 
then carried out by the exhaust and wasted. 

(c) The back pressure of the exhaust-steam, which depends 
on the form of the exhaust-passages, etc. This amounts to 
from 2 to 20% of the power developed. 

(d) Clearance-spaces. When cutting off at full stroke this 
waste is considerable (7 to 9%), but when the steam is used 
expansively the steam in these clearance-spaces expands and 
so its power is not wholly lost. 

(c) Radiation. In spite of all possible care in jacketing the 
cylinders, some heat is lost by radiation. 

(/) Radiation into the exhaust-steam. This is somewhat 
analogous to (h). Steam enters the cylinder at a temperature 
of, say, 361°; the walls of the cylinder are much cooler, say 250°; 
some heat is used in raising the temperature of the cylinder- 
walls; some steam is vaporized in so doing; when the exhaust 
is opened the temperature and pressure fall; the heat tem- 
porarily absorbed by the cjdinder-walls is reabsorbed by the 
exhaust-steam, re-evaporating the vapor previously formed, 
and thus a certain portion of heat-energy goes through the 
cylinder without doing any useful work. With an early cut-off 
the loss due to this cause is very great. 

The sum of all these losses is exceedingly variable. They 
are usually less at lower speeds. The loss in initial pressure 



378 RAILROAD CONSTRUCTION. § 322. 

(the difference between boiler pressure and the cylinder pres- 
sure at the beginning of the stroke) is frequently over 20%, 
but this is not all a net loss With an earl}^ cut-off the average 
cylinder pressure for the whole stroke is but a small part of 
the boiler pressure, yet the horse -power developed may be as 
great as, or greater than, that developed at a lower speed, later 
cut-off, and higher average pressure 

322. Tractive power The work done by the two cylinders 
during a complete revolution of the drivers evidentlj^ =area of 
pistons X average steam pressure X stroke X 2 X 2 . The resist- 
ance overcome evidently ^tractive force at circumference of 
drivers times distance traveled by drivers (which is the cir- 
cumference of the drivers) Therefore 

( area pistons X average steam pressure 

rr ^' x: < X stroke X 2X2. 

Tractive force = ) -. -^ .-r-. . 

{ circumference of drivers 

Dividing numerator and denominator by 7: (3.1415), we have 

r (diam piston) ^ X average steam 

rr^ J.' r < prcssurc X st rokc /-.o^^ 

Tractive force = J f, ^-^. , . (137) 

( diameter of driver 

which is the usual rule Although the rule is generally stated 
in this form, there are several deductions In the first place 
the net effective area of the piston is less than the nominal on 
account of the area of the piston-rod. The ratio of the areas 
of the piston-rod and piston varies, but the effect of this reduc- 
tion is usuallv from 1.3 to I '7%. No allowance has been made 
for friction — of the piston, piston-rod, cross-head, and the 
various bearings This would make a still further reduction 
of several per cent. Nevertheless the above simple rule is 
used, because, as will be shown, no great accuracy can be 
utilized. 

The tractive force is limited by the adhesion between the 
drivers and the rails, and this is a function of the weight on the 
drivers. Under the most favorable conditions this has been 
tested to amount to one-third the weight on the drivers, but 
such a ratio cannot be depended on Wellington used the 
ratio one-fourth The Baldwin Locomotive Works in their 
"Locomotive Data" give tables and diagrams based on J, ^^, 



§ 322. rolling-stock:. 379 

and 4 adhesion. As low a value as | or even | is occasionally 
used, but such a low rate of adhesion would only be found when 
the rails were abnormally slippery. In a well-designed loco- 
motive the tractive force, as computed above, and the tractive 
adhesion should be made about equal. The uncertainty in 
the coefficient of adhesion shows the futility of any refinement 
in the computation of the tractive force. 

It is only at very slow speeds that an engine can utilize all 
of its tractive force. When running at a high speed, the utmost 
horse-power that the engine can develop will only produce a 
draw-bar pull, which is but a small part of the possible tractive 
force. Power is the product of force times velocity. If the 
power is constant and the velocity increases, the force must 
decrease. This fact is well shown in the figures of some tests 
of a locomotive. The dimensions were as follows: cylinders, 
18"X24"; drivers, 68"; weight on drivers, 60000 lbs.; heating- 
surface, 1458 sq. ft.; grate-area, 17 sq. ft. During one test 
the average cylinder pressure was 83.3 lbs, (boiler pressure, 
145 ; 14-inch cut-off and throttle f open). By the above 
formula (137), 

^ ^. . 182X83.3X24 ^-^- „ 
Tractive force = — =9525 lbs. 

At i adhesion the tractive force was 15000 lbs; even at 4 ad- 
hesion, it would be 12000 lbs. This shows that at the speed 
of :his test (26.3 in. per hour) scarcely more than f of the trac- 
tive power was utilized. A still more marked case, shown by 
another test with the same engine, taken when the speed was 
53.4 miles per hour, indicated an average cylinder pressure of 
37.2 lbs., the throttle being ^ open and the valves cutting off 
at 8". In this case the tractive power, computed as before, 
equals 4254 lbs., about ^\ of the weight on the drivers and 
abo\it i of the tractive force which is possible at slow speeds. 
In the first case, the tractive power (9525) times the speed in 
feet per second (38.57) divided by 550 gives the indicated horse - 
powder, 668. In the second case, although the tractive force 
developed was so much less, the spe^ was much greater and 
the horse-power was about the same, 606. 

The above figures illustrate some of the foregoing statements 
regarding loss of efficiency. In both cases the steam was wire- 
dra\vTi. The boiler pressure was 145 lbs., but when the throttle 



380 RAILROAD COXSTRUCTION. *§ 322. 

was only | open and the steam was cut-off at 14'' (24" stroke) 
the average steam pressure in the cylinder was reduced to 
83.3 lbs. With the throttle but i open and the valves cutting 
off at 8'' (i of the stroke), the average pressure was cut down 
to 37.2 lbs. — about ^of the boiler pressure. Note that the heat- 
ing-surface per square foot of grate-area (1458-^17 = 86) is 
very large (see § 320) . Note also that the horse-power developed 
divided by the grate-area (17) gives 39 and 30 H.P. per square 
foot of grate-area. This is exceptionally large — 25 or 30 being 
a more common figure. 

The maximum tractive power is required when a train is 
starting, and fortunately it is at low velocities that the maxi- 
mum tractive force can be developed. The motion of the 
piston is so slow that there is but little reduction of steam 
pressure, and the valves are generally placed to cut off at full 
stroke. For the above engine, with 145 lbs. boiler pressure, 

^u u 1 . • c , ,' c ' 18^X145X24 

the absolute maximum ot tractive iorce is -^^ = 

o8 

16581 lbs. Of course, this maximum would never be reached 
unless the boiler pressure were increased. A common rule is 
to consider that the average effective. C3dinder pressure for slow 
speed and full stroke will be 80% of the boiler pressure. This 
would reduce the tractive force to the (nominal) value of 13265 
lbs., and the corresponding cylinder pressure would be 116 lbs. 
per square inch. With an effective cylinder pressure of about 
131 lbs. the tractive power is 15000 lbs., which is \ of the total 
weight on the drivers. This illustrates the general rule, stated 
above, that the cylinders, drivers, and boiler pressure should 
be so proportioned that the maximum tractive force should 
about equal the maximum adhesion which could be obtained. 

As another numerical example, the dimensions of a recently 
constructed heavy consolidation engine are quoted. The cylin- 
ders are 24''X32"; diameter of drivers, 54"; total w^eight of 
engine and tender, 391400 lbs.; weight of engine, 250300 lbs.; 
weight on drivers, 225200 lbs.; capacity of tender, 7500 gallons; 
the boiler has 406 tubes, 2^" in diameter and 15' long; fire- 
box, 132"X40i"; heating-surface of tubes, 3564 sq. ft.; of 
fire-box, 241 sq. ft. — total, 3805 sq. ft.; boiler pressure, 220 lbs. 
per square inch. Applying Eq. 132, we may compute 75093 
lbs. as the absolute maximum of tractive power. In fact this 
is an unattainable limit, for reasons before stated. The trac- 



§ 323. ROLLING-STOCK. 381 

tive force is given as 63000, which corresponds to an effective 
cyhnder pressure of about 185 lbs., about 84% of the boiler 
pressure. This tractive force is 28% of the weight on the 
drivers, a tractive ratio of 1 : 3.6. 

RUNNING GEAR. 

323. Types of running gear, (a) "American." This was 

Q/^ once the almost universal type for 
k— Z Q Q -^ both passenger and freight service. 

It is still very commonly used for passenger service, but it is 
not the best form for heavy freight work. 

(b) "Columbia." Four drivers, one pair of pilot-truck w^heels 
and one pair of trailing wheels be- /^>. /^^ -. 

hind the drivers. The low trailing ^ \^ k-^ — -^ -^ 

wheels permit a desirable enlargement of the fire-box. This 
is a recent type, used exclusively for passenger service. 

Q-^ (c) "Atlantic." Similar to 
LJ Q Q -^ h except that the pilot-truck 

has four wheels instead of two. 

(d) "Mogul." These are used for both passenger and freight 
service, but are not well /^ /^^ ^^~^ 

adapted for either high speed v ^ v >^ V >^ O 

or great tractive power. 

(e) "Ten-wheel." Similar to d except that the pilot-truck 

Q^-^ >^-v has four wheels instead of 

V J V y 00 "^ two. The use is similar to 
that of d. 

(f) "Consolidation." The present standard for freight ser- 
vice. It permits great trac- /^^ /^^ /^^ /'^ 

tive power without excessive v ^ \. y \. y \ y Q -^ 

concentrated loads on the track. 

(g) Switching-engines. These have four or six (and excep- 
tionally even eight or ten) drivers and no truck-wheels. They 
are only adapted for slow^ speed when a maximum of tractive 
power is needed. Sometimes the water-tank and even a small 
fuel-box is loaded on. Since fuel is always near at hand for a 
yard -engine, the fuel-box need not be large. 

(h) " Double-en ders." As explained in § 315, truck-wheels are 
needed in front of the drivers to guide them around curves. If 
an ordinary engine is run backward, the flanges of the rear 



382 RAILROAD CONSTRUCTION. § 324. 

drivers will become badly worn, and if the speed is high, the 

danger of derailment is considerable. In suburban service, 

-^ /^ /^ _^ when the runs are short, it is 

y — ^^.Z X_-Z ^ preferable to run the engines 

forward and backward, rather than turn them at each end of 
the route. Therefore a pilot-truck is placed at each end. 

(i) " Miscellaneous types." Almost every conceivable com- 
bination of drivers and truck-wheels has been used. The 
"Mastodon" is similar to the ''Consolidation" except that the 
pilot-truck has four w^heels instead of two. The "Decapod" 
has ten driving-wheels. The "Forney" (named after the in- 
ventor) has been very extensively used on elevated roads. The 
weight of the boiler and machinery is carried on four driving- 
wheels; the engine-frame is extended so as to include a small 
tank and fuel-box, the weight of which is chiefly supported by 
a truck of two or four wheels. They run best when running 
"backward," i.e., tender first. 

The great variation in types of running gear which has been 
developed in recent years, has started a convenient and unmis- 
takable method of indicating the running gear. Commencing 
with the front of the engine (or pilot) always at the left (instead 
of at the right, as in the illustrations above) the number of 
wheels of the pilot truck on both rails is indicated by 0, 2 or 4, 
according as there is no pilot truck, a two- wheeled or a four- 
w^heeled pilot truck. Then the number of drivers on both rails 
is indicated by the next number and the number of trailing 
wheels by the third number. The running gear of the tender is 
not indicated. This method may be illustrated by applying it 
to the types indicated above: 

American 4-4-0 

Columbia 2-4-2 

Atlantic 4-4-2 

Mogul 2-6-0 

Ten-wheel 4-6-0 

Consolidation 2-8-0 

Six-wheel switcher 0-6-0 

Mastodon 4-8-0 

The running gear of any new type may thus be unmistakably 
indicated by three figures. 



§ 324. ROLLING-STOCK. 383 

324. Equalizing-levers. The ideal condition of track, from 
the standpoint of smooth running of the rolling stock, is that 
the rails should always lie in a plane surface. While this con- 
dition is theoretically possible on tangents, it is unobtainable 
on curves, and especially on the approaches to curves when the 
outer rail is being raised. Even on tangents it is impossible 
to maintain a perfect surface, no matter how perfectly the 
track may have been laid. In consequence of this, the points 
of contact of the wheels of a locomotive, or even of a four- 
wheeled truck, will not ordinarily lie in one plane. The rougher 
and more defective the track, the worse the condition in this 
respect. Since the frame of a locomotive is practically rigid, 
if the frame rests on the driver-axles through the medium of 
springs at each axle-bearing, the compression of the springs 
(and hence the pressure of the drivers on the rail) will be varia- 
ble if the bearing-points of the drivers are not in one plane 
This variable pressure affects the tractive power and severely 
strains the frame. Applying the principle that a tripod will 
stand on an uneven surface, a mechanism is employed which 
virtually supports the locomotive on three points, of which one 
is usually the center-bearing of the for^^ard truck. On each 
side the pressure is so distributed among the drivers that even 
if a driver rises or falls with reference to the others, the load 
carried by each driver is unaltered, and that side of the engine 
rises or falls by one nth of the rise or fall of the single driver, 
where n represents the number of wheels. The principle in- 
volved is shown in an exaggerated form in Fig. 195. In the 
diagram, MN represents the normal position of the frame when 
the wheels are on line. The frame is supported b}^ the hangers 
at a, Cj /, and h. ah, de, and gh are horizontal levers vibrating 
about the points H, K, and L, which are supported by the 
axles. While it is possible with such a system of levers to make 
MN assume a position not parallel with its natural position, 
yet, by an extension of the principle that a beam balance loaded 
with equal weights will always be horizontal, the effect of rais- 
ing or lowering a wheel will be to move MA^ parallel to itself. 
It only remains to determine how much is the motion of MN 
relative to the rise or drop of the wheel. 

The dotted lines represent the positions of the wheels and 
levers when one wheel drops into a depression. The wheel 
center drops from p to q, a, distance m. L drops to U, a 
distance m (see Fig. 195, b) ; M drops to Af' , an unknown dis- 



384 



KAILROAI) CONSTRUCTION. 



§324. 



tance x; therefore aa' =x\ by = x; cc'=x; dd' = ^x = ee'] ff =x) 
.-. gg' = 5.r ; hh' =x; LU = i(gg' + hh') = }(6.t) =777 ; .\x = lm\ 
i.e., MN drops, parallel to itself, 1/n as much as the wheel 
drops, where n is the number of wheels. The resultant effect 
caused by the simultaneous motion of U\o wheels with refer- 

1L 




7n^~~cv 



Fig. 195. — Action of Equalizing-levers. 

ence to the third is evidently the algebraic sum of the effects 
of each wheel taken separately. 

The practical benefits of this device are therefore as follows: 

(a) When any driver reaches a rough place in the track, a 
high place or a low place, the stress in all the various hangers 
and levers is unchanged. 

(6) The motion of the frame (represented by the bar MN 
in Fig. 195) is but 1/n of the motion of the wheel, and the jar 
and vibration caused b}^ a roughness in the track is correspond- 
ingly reduced. 

The details of applying these principles are varied, but in 
general it is done as follows; 

(a) American and ten-wheeled types. Drivers on each side 
form a system. The center-bearing pilot-truck is the third 
point of support. The method is illustrated in Fig. 196. 

(b) Mogul and consolidation types. The front pair of drivers 
is connected with the two-wheeled pilot-truck (as illustrated 
in Fig. 197) to form one system. The remaining drivers on 
each side are each formed into a system 

The de\dce of equalizers is an American invention. Until 
recently it has not been used on foreign locomotives. The 
necessitv for its use becomes less as the track is maintained 



§324. 



ROLLING-STOCK. 



385 



with greater perfection and is more free from sharp curves. 
A locomotive not equipped with this device would deteriorate 





very rapidly on the comparatively rough tracks which are 
usually found on light-traffic roads. It is still an open ques- 



386 RAILROAD CONSTRUCTION. § 325. 

tion to what extent the neglect of this device is responsible 
for the statistical fact that average freight-train loads on foreign 
trains are less in proportion to the weight on the drivers than 
is the case with. American practice. The recent increasing use 
of this device on foreign heavy freight locomotives is perhaps 
an acknowledgment of this principle. 

325. Counterbalancing. At very high velocities the cen- 
trifugal force developed by the weight of the rotating parts 
becomes a quantity which cannot be safely neglected. These 
rotating parts include the crank-pin, the crank-pin boss, the 
side rod, and that part of the weight of the connecting-rod 
which may be considered as rotating about the center of the 
crank-driver. As a numerical illustration, a driving-wheel 
62" in diameter, running 60 miles per hour, will revolve 325 
times per minute. The weights are: 

Crank-pin 110 lbs. 

boss 150 '' 

One-half side rod 240 '' 

Back end of connecting-rod 190 *' 

Total 690 lbs. 

If the stroke is 24", the radius of rotation is 12", or 1 foot. Then 
Gv' 690X47:^2X3252 _^_.. „ 
-gf- 32.2X1X60- =248211bs., 

which is half as much again as the weight on a driver, 16000 lbs. 
Therefore if no counterbalancing were used, the pressure be- 
tween the drivers and the rail would always be less (at any 
velocity) when the crank-pin was at its highest point. At a 
velocity of about 48 miles per hour the pressure would become 
zero, and at higher velocities the wheel would actually be 
thrown from the rail. As an additional objection, when the 
crank-pin was at the lowest point, the rail pressure would be 
increased (velocity 60 miles per hour) from 16000 lbs. to nearly 
41000 lbs., an objectionably high pressure. These injurious 
effects are neutralized by ^^counterbalancing." Since all of 
the above-mentioned weights can be considered as concen- 
trated at the center of the crank-pin, if a sufficient weight is so 
placed in the drivers that the center of gravity of the eccentric 
weight is diametrically opposite to the crank-pin, this centrifu- 
gal force can be wholly balanced. This is done by filhng up 
a portion of the space between the spokes. If the center of 
gravity of the counterbalancing weight is 20" from the center, 



§ 325. ROLLING STOCK. 387 

then, since the crank-pin radius s 12", the required weight 
would be 690 X II = 414 lbs. 

In addition to the effect of these revolving parts there is 
the effect of the sudden acceleration and retardation of the 
reciprocating parts. In the engine above considered the weights 
of these reciprocating parts will be: 

Front end of connecting-rod 150 lbs. 

Cross-head. ." 174 ' * 

Piston and piston-rod 300 '^ 

Total 624 lbs. 

Assume as before that the reciprocating parts may be con- 
sidered as concentrated at one point, the point P of the dia- 
gram in Fig. 198. Since the 

UiOtion of P is horizontal ,^''' ""^W' 

oiily, the force required to / /''''^j---^^ \ 

overcome its inertia at any - ■ ' i X tX « 




point will exactly equal ^^ ' ^ — *^i^_\Vi^ 

the horizontal component of '\S^ "^ 

the force required to over- ^^-^^ ^^X 

come the inertia of an equal 

weight at S revolving in ^^^- 198.-Action of Co ntehbalance. 

a circular path. Then evidently the horizontal component of 
the force required to keep TT^ in the circular path will exactly 
balance the force required to overcome the inertia of P. Of 
course TT^ = P. But a smaller weight TT^', whose weight is 
inversely proportional to its radius of rotation, wdll evidently 
accomplish the same result. In the above numerical case, if 
the center of gravity of the counterweights is 20" from the 
center, the required weight to completely counterbalance 
the reciprocating parts would be 624 X^f = 374.4 lbs. This 
counterweight need not be all placed on the driver carrying 
the main crank-pin, but can be (and is) distributed among all 
the drivers. Suppose it w^ere divided between the two drivers 
in the above case. At 60 miles per hour such a counterweight 
w^ould produce an additional pressure of 11211 lbs. when the 
counterw^eight was dowm, or a lifting force of the same amount 
when the counterweight was up. Although this is not suffi- 
cient to lift the driver from the rail, it would produce an objec- 
tionably high pressure on the rail (over 27000 lbs.), thus inducing 
just what it was desired to avoid on account of the eccentric 
rotating parts. Therefore a compromise must be made. Only 
a portion (one half to three fourths) of the weight of the recip- 
rocating parts is balanced. Since the effect of the rotating 



388 RAILROAD CONSTRUCTION. § 325. 

weights is to cause variable pressure on the rail, while the effect 
of the reciprocating parts is to cause a horizontal a\ obnling or 
^'nosing'' of the locomotive, it is impossible to balance both. 
Enough counterweight is introduced to partially neutralize the 
effect of the reciprocating parts, still leaving some tendency 
to horizontal wobbling, while the counterweights which Avere 
introduced to reduce the wobbling cause some variation oi 
pressure. By using hollow piston-rods ^of steel, ribbed cross- 
heads, and connecting- and side-rods with an I section, the 
weight of the reciprocating parts may be greatly lessened with- 
out reducing their strength, and with a decrease in weight the 
effect of the unbalanced reciprocating parts and of the '' excess 
balance" (that used to balance the reciprocating parts) is 
largely reduced. 

Current practice is somewhat variable on three features: 

(a) The proportion of the weight of the connecting-rod which 
should be considered as revolving weight. 

(b) The proportion of the total reciprocating weight that 
should be balanced. 

(c) The distribution among the drivers of the counterweight 
to balance the reciprocating parts. 

An exact theoretical analysis of (a) shows that it is a func- 
tion of the weights and dimensions of the reciprocating parts. 
The weight which may be considered as revolving equals * 






r' + k 

in which r = radius of the crank, Z = length of connecting-rod, 
A; = distance of center of gyration from wrist-pin, (i = distance 
of center of gravity from wrist-pin, M^i= weight of connecting- 
rod in pounds, and 11^2 = weight of piston, piston-rod, and cross- 
head in pounds; all dimensions in feet. An application of this 
formula will show that for the dimensions of usual practice, 
from 51 to 57% of the weight of the connecting-rod should be 
considered as revolving weight. 

The principal rules which have been formulated for counter- 
balancing may be stated as follows: 

1. Each w^heel should be balanced correctly for the revolving 
parts connected with it. 

2. In addition, introduce counterbalance sufficient for 50% 
of the weight of the reciprocating parts for ordinary engines, 

*R. A. Parke, inR. RTCazette, Feb. 23, 1894. 



§ 326. 



ROLLING-STOCK. 



389 



increasing this to 75% when the reciprocating parts are exces- 
sively heavy (as in compound locomotives) or when the engine 
is light and unable to withstand much lateral strain or when 
the wheel-base is short. 

3. Consider the weight of the connecting-rod as h revolving 
and i reciprocating when it is over 8 feet long; when shorter 
than 8 feet, consider -f^ of the w^eight as revolving and j% as 
reciprocating. 

4. The part of the weight of the connecting-rod considered 
as revolving should be entirely balanced in the crank-driver wheel. 

5. The ^^ excess balance^' should be divided equally among 
the drivers. 

6. Place the counterbalance as near the rim of the wheel 
as possible and also as near the outside 
of the wheel as possible in order that 
the center of gravity shall be as near 
as possible opposite the center of 
gravity of the rods, etc., w^liich are all 
outside of even the plane of the face 
of the wheel. 

In Fig. 199 is show^n a section of a 
locomotive driver with the cavities in 
the casting for the accommodation of 
the lead which is used for the counter- 
balance weight. Incidentally several 
other features and dimensions are showTi 
in the illustration. 

326. Mutual relations of the boiler power, tractive power, 
and cylinder power for various types. The design of a locomo- 
tive includes three distinct features which are varied in their 
mutual relations according to the w^ork which the engine is 
expected to do. 

(a) The boiler power. This is limited by the rate at which 
steam may be generated in a boiler of admissible size and weight. 
Engines which are designed to haul very fast trains w^hich are 
comparati^^ely light must be equipped with very large grates and 
heating surfaces so that steam may be developed Avith great 
rapidity in order to keep up with the very rapid consumption. 
Engines for very heavy freight work are run at very much 
low^er velocity and at a lower piston speed in spite of the fact 
that more strokes are required to cover a given distance and 
the demand on the boiler for rapid steam production is not 




Fig. 199. — Section of 
Locomotive-driver. 



390 RAILROAD CONSTRUCTION. § 326. 

as great as with high-speed passenger-engines. The capacity of 
a boiler to produce steam is therefore Hmited by the hmiting 
weight of the general type of engine required. Although im- 
provements may be and have been made in the design of fire- 
boxes so as to increase the steam-producing capacity without 
adding proportionately to the weight, yet there is a more or 
less definite limit to the boiler power of an engine of given 
weight. 

(b) The tractive power. This is a function of the weight on 
the drivers. The absolute limit of tractive adhesion between a 
steel-tired wheel and a steel rail is about one third of the pressure, 
but not more than one fourth of the weight on the drivers can 
be depended on for adhesion and wet rails will often reduce 
this to one fifth and even less. The tractive power is therefore 
absolutely limited by the practicable weight of the engine. In 
some designs, when the maximum tractive power is desired, not 
only is the entire weight of the boiler and running gear thrown 
on the drivers, but even the tank and fuel-box are loaded on. 
Such designs are generally employed in switching-engines (or 
on engines designed for use on abnormally heavy mountain 
grades) in which the maximum tractive power is required, but 
in which there is no great tax on the boiler for rapid steam pro- 
duction (the speed being always very low), and the boiler and 
fire-box, which furnish the great bulk of the weight of an engine, 
are therefore comparatively light, and the requisite weight for 
traction must, therefore, be obtained by loading the drivers 
as much as possible. On the other hand, engines of the highest 
speed cannot possibly produce steam fast enough to maintain 
the required speed unless the load be cut down to a compara- 
tively small amount. The tractive power required for this 
comparatively small load will be but a small part of the weight 
of the engine, and therefore engines of this class have but a 
small proportion of their weight on the drivers; generally 
have but two driving-axles and sometimes but one. 

(c) Cylinder power. The running gear forms a mechanism 
which is simply a means of transforming the energy of the boiler 
into tractive force and its power is unlimited, within the prac- 
tical conditions of the problem. The power of the running 
gear depends on the steam pressure, on the area of the piston, 
on the diameter of the drivers, and on the ratio of crank-pin 
radius to wheel radius, or of stroke to driver diameter. It 



§ 326. 



ROLLING-STOCK. 



391 



is alwaj's possible to increase one or more of these elements 
by a relatively small increase of expenditure until the cylinders 
are able to make the drivers slip, assuming a sufficiently great 
resistance. Since the power of the engine is limited by the 
power of its weakest feature, and since the running gear is the 
most easily controlled feature, the power of the running gear 
(or the ''cylinder power") is always made somewhat excessive 
on all well-designed engines. It indicates a badly designed 
engine if it is stalled and unable to move its drivers, the steam 
pressure being normal. If it is attempted to use a freight- 
engine on fast passenger service, it will probably fail to attain 
the desired speed on account of the steam pressure falling. 
The tractive power and cylinder power are superabundant, but 
the boiler cannot make steam as fast as it is needed for high 
speed, especially w^hen the drivers are small. The practical 
result would be a comparatively low speed kept up with a forced 
fire. If it is attempted to use a high-speed passenger-engine 
on heavy freight service, the logical result is a slipping of the 
drivers until the load is reduced. The boiler power and cylinder 
power are ample, but the weight on the drivers is so small that 
the tractive power is only sufficient to draw a comparatively 
small load. 

These relations between boiler, C3dinder, and tractive power 
are illustrated in the following comparative figures referring 
to a fast passenger-engine, a heavy freight-engine, and a switch- 
ing-engine. The weights of the passenger- and freight-engines 
are about the same, but the passenger-engine has only 72% of 



Kind. 


Cylinders. 


Total 
Wght. 


Wt. on 
Driv'rs 


Heat- 
ing 
Sur- 
face, 

sq. ft. 


Grate 
area 
sq. ft. 


Steam 
Pres- 
sure in 
Boiler. 


Stroke. 
Diam. 
Driver. 


Fast passenger . 
Heavy freight . 
Switcher 


19''X24'' 

20"X24'' 
19''X24'' 


126700 
128700 
109000 


81500 
112600 
109000 


1831.8 
1498.3 
1498.0 


26.2 
31.5 

22.8 


180 
140 
160 





the tractive power of the freight. But the passenger-engine 
has 22% more heating-surface and can generate steam much 
faster: it makes less than two thirds as many strokes in cover- 
ing a given distance, but it nms at perhaps twice the speed 



392 RAILROAD CONSTRUCTION. § 326, 

and probably consumes steam much faster. The switch- 
engine is Hghter in total weight, but the tractive power is nearly 
as great as the freight and much greater than the passenger- 
engine. While the heating-surfaces of the freight- and switch- 
ing-engines are practically identical, the grate area of the switcher 
is much less; its speed is always low and there is but little neces- 
sity for rapid steam development. 

While these figures show the general tendency for the relative 
proportions, and in this respect may be considered as typical, 
there are large variations. The recent enormous increase in 
the dead weight of passenger-trains has necessitated greater 
tractive power. This has been provided sometimes by using 
^'MoguP^ and ^Hen-wheel'^ engines, which were originally 
designed for freight work. On the other hand, the demand 
for fast freight service, and the possibility of safely operating 
such trains by the use of air-brakes, has required that heavy 
freight-engines shall be run at comparatively high speeds, and 
that requires the rapid production of steam, large grate areas, 
and heating surfaces. But in spite of these variations, the 
normal standard for passenger service is a four-driver engine 
carr^^ing about two thirds of the weight of the engine on the 
drivers, Avhich are very large; the normal standard for freight 
work is the ^'consolidation," with perhaps 90% of the weight 
on the drivers, which are small, but which must have the pony 
truck for such speed as it uses; and finally the normal standard 
for switching service has all the weiglit on the drivers and has 
comparatively low steam-producing capacity. 

327. Life of locomotives. The life of locomotives (as a 
whole) may be taken as about 800000 miles or about 22 to 24 
3^ears. While its life should be and is considered as the period 
between its construction and its final consignment to the scrap 
pile, parts of the locomotive may have been renewed more 
than once. The boiler and fire-box are especially subject to 
renewal. The mileage life is much longer than formerl}'. This 
is due partly to better design and partly to the custom of 
drawing the fires less frequently and thereby avoiding some 
of the destructive strains caused by extreme alterations of 
heat and cold. Recent statistics give the average annual 
mileage on twenty-three leading roads to be 41000 miles. 



§ 328. ROLLING-STOCK. 393 



328. Capacity and size of cars. The capacity of freight-cars 

has been enormously increased of late years. About thirty 
years ago the usual live-load capacity for a box-car was about 
20000 lbs. In 1893 the standard box-car, gondola-cars, etc., 
of the Pennsylvania Railroad on exhibition at the Chicago 
Exposition, had a live-load capacity of 60000 lbs. and a dead 
weight of 30000 to 33000 lbs With a full load, the w^eight on 
each wheel is nearly 12000 lbs , which equals or exceeds the 
load usually placed on the drivers of ordinary locomotives. 
But now cars with a live-load capacity of 80000 lbs. are almost 
standard, 100000-lb. cars are very common, and even larger cars 
are made for special service. (See Fig. 200.) 

The limitation of the carr\ang capacity for some kinds of 
freight depends somewhat on the amount of live load that 
can be carried within given dimensions; for the cross-section of 
a car is limited to the extreme dimensions which may be safely 
run through the tunnels and through bridges as at present 
constructed, and the length is somew^hat limited by the dif- 
ficulty of properly supporting an excessively hesivy load, dis- 
tributed over an unusually long span, by a structure w^hich is 
subjected to excessive jar, concussion, compression, and ten- 
sion. The cross-sectional limit seems to have been scarcely 
reached yet, except, perhaps, in the case of furniture and carriage- 
cars, whose load per cubic foot is not great. The usual width 
of freight-cars is about 8 to 9 feet, w^hile parlor-cars and sleepers 
are generally 10 feet wdde and sometimes 11 feet. The highest 
point of a train is usually the smoke-stack of the locomotive 
which is generally 14 feet above the rails and occasionally over 
15 feet. A sleeping-car usually has the highest point of the 
car about 14 feet above the rails. Box-cars are usually about 
8 feet high (above the sills), with a total height of about 11' 3". 
Refrigerator-cars are usually about 9' high and furniture-cars 
about 10' above the sills, the truck adding about 3' 3". The 
usual length is 34 feet, but 35 to 40 feet is not uncommon. 
Passenger-cars (day coaches) are usually 50 feet long, exclusive of 
the end platforms and weigh 45000 to 50000 lbs. Sixty pas- 
sengers at 150 pounds apiece (a high average) will only add 
9000 lbs. to the weight. A parlor-car or sleeper is generally 
about 65 feet long exclusive of the platforms, which add about 
6' 0". The weight is an\-^'here from 60000 to 80000 lbs 



394 RAILROAD CONSTRUCTION. § 329. 

The weight of the 25 or 30 passengers it may carry is hardly 
worth considering in comparison. 

329. Stresses to which car-frames are subjected. A car 
is structurally a truss, supported at points at some distance 
from the ends and subjected to transverse stress. There is, 
therefore, a change of flexure at two points between the trucks. 
Besides this stress the floor is subjected to compression when 
the cars are suddenly stopped and to tension when in ordinary 
motion, the tension being greater as the train resistance is 
geater and as the car is nearer the engine. The shocks, jars, 
and sudden strains to which the car-frames are subjected are 
veiy much harder on them than the mere static strains due to 
their maximum loads if the loads were quiescent. Consequently 
any calculations based on the static loads are practically value- 
less, except as a very rough guide, and previous experience 
must be relied on in designing car bodies. As evidence of the 
increasing demand for strength in car-frames, it has been re- 
cently observed that freight-cars, built some years ago and 
built almost entirely of wood, are requiring repairs of wooden 
parts which have been crushed in service, the wood being per- 
fectly soimd as regards decay. 

330. The use of metal. The use of metal in car construction 
is very rapidly increasing.. The demand for greater strength 
in car-frames has grown until the wooden framing has become 
so heavy that it is found possible to make steel frames and 
trucks at a small additional cost, the steel frames being twice 
as strong and yet reducing the dead weight of the car about 
5000 lbs., a consideration of no small value, especially on roads 
having heavy grades. Another reason for the increasing use 
of metal is the great reduction in the price of rolled or pressed 
steel, while the cost of wood is possibly higher than before. 
The advocates of the use of steel ad\4se steel floors, sides, etc. 
For box-cars a wooden floor has advantages. For ore and 
coal-cars an all-metal construction has advantages. (Fig. 200.) 
In Germany, where steel frames have been almost exclusively 
in use for many years, they have not yet been able to determine 
the normal age limit of such frames; none have yet worn out. 
The life is estimated at 50 to 80 years. 




lOOOOO-LB. Box Car. 




Steel Coal Car. 




Wooden Box Car; Steel Frame.? 
Fig. 200. — Some Heavy Freight Cars. 
{To face page 394.) 



§ 331. ROLLING-STOCK* 395 

Brake beams are also best made of metal rather than wood, 
as was formerly done. Metal brake-beams are generally used on 
cars having air-brakes, as a wooden beam must be excessively 
large and heavy in order to have sufficient rigidity. 

Truck-frames (see Fig. 201), which were formerly made prin- 
cipally of wood, are now largely made of pressed steel. It makes 




Fig. 201. 

a reduction in weight of about 3000 lbs. per car. The increased 
durability is still an uncertain quantity. 

331. Draft gear. The enormous increase in the weight and 
live load capacities of rolling stock have necessitated a corre- 
sponding development in draft gear. Even within recent years, 
"coal- jimmies,'' carrying a few tons have been made up into 
trains by dropping a chain of three big links over hooks on the 
ends of the cars. But the great stresses due to present loadings 
would tear such hooks from the cars or tear the cars apart if 
such cars were used in the make-up of long heavy trains as now 
operated. The next stage in the development of draft gear was 
the invention of the ''spring coupler,'' by which the energy due 
to a sudden tensile jerk or the impact of compression may be 
absorbed by heavy springs and gradually imparted to the car 
body. Such devices, for which there are many designs, seemed 
to answer the purpose for cars of 25 to 40 tons capacity. The 
use of 100,000-pound steel cars soon proved the inadequacy of 
even spring couplers. The friction-draft gear was then in- 
vented. The general principle of such a gear is that, when 



396 



EAILROAD CONSTRUCTION. 



§331. 





§ 332. ROLLING-STOCK. 397 

acting at or near its maximum capacity, it harmlessly trans- 
forms into heat the excessive energy developed by jerks or 
compression. There are several different designs of such gear, 
but the general principle underlying all of them may be illus- 
trated by a description of the Westinghouse draft gear. The 
gear employs springs which have sufficient stiffness to act as 
ordinary spring-couplers for the ordinary pushing and pulling , 
of train operations. Sections of the gear are shown in Fig. 202, 
while the method of its application to the framing of a car of 
the pressed steel type is shown in Fig. 203, a and h. When 
the draft gear is in tension the coupler, which is rigidly attached 
to B, is drawn to the left, drawing the follower Z with it. Com- 
pression is then exerted through the gear mechanism to the 
follower A which, being restrained by the shoulders RR^ against 
which it presses, causes the gear to absorb the compression. 
The coil-spring C forces the eight wedges n against the eight 
corresponding segments E, The great compression of these 
surfaces against the outer shell produces a friction w^hich retards 
the compression of the gear. The total possible movement of 
the gear, as determined by an official test, was 2.42 inches, when 
the maximum stress was 180,000 pounds. The work done in 
producing this stress amounted to 18,399 foot-pounds. Of this 
total energy 16,1566 foot-pounds, or over 90%, represents the 
amount of energy absorbed and dissipated as heat by the 
frictional gear. The remaining 10% is given back by the 
recoil. The main release spring K is used for returning the 
segments and friction strips to their normal position after the 
force to close them has been removed. It also gives additional 
capacity to the entire mechanism. The auxiliary spring L 
releases the wedge D, while the release pin M releases the pres- 
sure of the auxiliary spring L against the wedge during fric- 
tional operation. If we omit from the above design the fric- 
tional features and consider only the two followers A and Z, 
separated by the springs C and K, acting as one spring, we have 
the essential elements of a spring-draft gear. In fact, this 
gear acts exactly like a spring-draft gear for all ordinary service, 
the frictional device only acting during severe tension and com- 
pression. 

332. Gauge of wheels and form of wheel-tread. — In Fig. 204 
is shown the standard adopted by the Master Car Builders' 
Association at their twentieth annual convention. Note the 



398 



RAILROAD CONSTRUCTION. 



§ 332. 






-^ 












O 




o 
o 




( 


) 


- 1 












" ■! 


) 


i\> i 




O 
( 


n n 


) 






( 
( 




> 
) 








N 

T T 


] p 






u f" 






G . 






^ CO p 


a 






c 
( 


< 

ni o p 








( 


) ' ' ( 


) 






( ) 


nl n 


) 






( ) 


( 


) 




,= 


°1 


^.A.^ 








§ 333. ROLLING-STOCK. 399 

normal position of the gauge-line on the wheel-tread. In 
Fig. 114, p. 238, the relation of rail to wheel-tread is shown 
on a smaller scale. It should be noted that there is no definite 
position Avhere the wheel-flange is absolutely '^ chock-a-block'' 
against the rail. As the pressure increases the wheel mounts 
a little higher on the rail until a point is soon reached when the 
resistance is too great for it to mount still higher. By this 
means is avoided the shock of unyielding impact w^hen the car 
sways from side to side. When the gauge between the inner 
faces of the wheels is greater or less than the limits given in 
the figure, the interchange rules of the Master Car Builders' 
Association authorize a road to refuse to accept a car from 
another road for transportation. At junction points of rail- 
roads inspectors are detailed to see that this rule (as well as 
many others) is complied with in respect to all cars offered 
for transfer. 

TRAIN-BRAKES. 

333. Introduction. Owing to the very general misappre- 
hension that exists regarding the nature and intensity of the 
action of brakes, a complete analysis of the problem is con- 
sidered justifiable. This misapprehension is illustrated by the 
common notion (and even practice) that the effectiveness of 
braking a car is' proportional to the brake pressure, and there- 
fore a brakeman is frequently seen using a bar to obtain a 
greater leverage on the brake-wheel and using his utmost 
strength to obtain the maximum pull on the brake-chain while 
the car is skidding along with locked wheels. 

When a vehicle is moving on a track with a considerable 
velocity, the mass of the vehicle possesses kinetic energy of 
translation and the wheels possess kinetic energy of rotation. 
To stop the vehicle, this energy must be destroyed. The 
rotary kinetic energy will vary from about 4 to 8% of the 
kinetic energy of translation, according to the car loading 
(see § 347). On steam railroads brake action is obtained by 
pressing brake-shoes against car-wheel treads. As the brake- 
shoe pressure increases, the brake-shoes retard with increasing 
force the rotary action of the wheels. As long as the wheels 
do not slip or ^'skid" on the rails, the adhesion of the rails 
forces them to rotate with a circumferential velocity equal to 
the train velocity. The retarding action of the brake-shoe 



400 



RAILROAD CONSTRUCTION. 



I 333. 




Fig. 204. — M. C. B. Standard Wheel-tread and Axle. 



§ 334. ROLLING-STOCK. 401 

checks first the rotative kinetic energy (which is small), and 
the remainder develops a tendency for the wheel to slip on the 
rail. Since the rotative kinetic energy is such a small per- 
centage of the total, it wdll hereafter be ignored, except as 
specifically stated, and it will be assumed for simplicity that 
the only work of the brakes is to overcome the kinetic energy 
of translation. The possible effect of grade in assisting or 
preventing retardation, and the effect of all other track resist- 
ances, is also ignored. The amount of the developed force 
which retards the train movement is limited to the possible 
adhesion or static friction between the wheel and the rail. 
When the friction between the brake-shoe and the wheel ex- 
ceeds the adhesion between the wheel and the rail, the wheel 
skids, and then the friction betw^een the w^heel and the rail 
at once drops to a much less quantity. It must therefore be 
remembered at the outset that the retarding action of brake- 
shoes on wheels as a means of stopping a train is absolutely 
limited by the possible static friction between the braked 
wheels and the rails. 

334. Laws of friction as applied to this problem. Much of 
the misapprehension regarding this problem arises from a very 
common and widespread misstatement of the general law^s of 
friction. It is frequently stated that friction is independent 
of the velocity and of the unit of pressure. The first of these 
so-called law^s is not even approximately true. A very exhaus- 
tive series of tests were made by Capt. Douglas Gait on on the 
Brighton Railw^ay in England in 1878 and 1879, and by M. 
George Marie on the Paris and Lyons Railw^ay in 1879, with 
trains which were specially fitted with train-brakes and w ith 
dynagraphs of various kinds to measure the action of the 
brakes. Experience proved that variations in the condition of 
the rails (w^et or dry), and numerous irregularities incident to 
measuring the forces acting on a heavy body moving with a 
high velocity, were such as to give somewhat discordant re- 
sults, even w^hen the conditions w^ere made as nearly identical 
as possible. But the tests w^ere carried so far and so persist- 
ently that the general law^s stated below were demonstrated 
beyond question, and even the numerical constants w^ere deter- 
mined as closely as they may be practically utilized. These 
laws may be briefly stated as foUow^s: 

(a) The coefficient of friction betw^een cast-iron brake-blocks 



^^2 RAILROAD CONSTRUCTION. \ 

and steel tires is about .3 when the wheels are "iu> 
mg ; ,t drops to about .16 when the velocity is Tbout ' 
per hour and is less than .10 when the velodtv is Sn' 
onLrl^lftTdrJ"^^-- --^--'>' --"e c;; 

q e t?of It: fTf T"" °^ *^'^ '- wreTabI shed 
question, the tests to demonstrate the law of the varial 

.fit: :s:^-i^j^r^ic^,t^:;^ 

rail-sometimes less than one third as much ^ ' 

(^ An analysis of the tests aU pointed' to a law th 
faction developed does not increase as rapidly JZ 
of pressure increases, but this ma.. L ii u ''^ 

an established law. ^ '"'"''^'y ''" ''°°«ider, 

(e) The adhesion between the wheel and th^ . -i 

be independent of ^-elocitv The IT u '' ^^^^ 

that must be developed beforJS I'T ^""^ '"^^'^^ '^' 

The practical effecTof £" trtlh"" 1^ °". *'^ "' 

observed phenomena- '^°"'" "^^ t'^e foU«' 

the wheel hfs a velhth I^ r T '' '^"^ ^'^ ^^'^ f^*^* 
as the square of tl^ir'^J^'f' '"^''"^ ^^-'^'^'^ - 
first), but it is chll ? ^ f^ ^'^'^'^ """^t be o^erc 

frJ;n at th higher' vebciW f ' '^'^^ ^'^^^ *^^ -^«"- 
hour it is about .ofrwMe hLdh"'^ T'" ^"' ''^ '"''^^ 
the rail is independir!:;^^""" '^"^" ^'^ ^"'^^^^ 
deLLferi\rJ°l^els;^^^^^^^^^^^^^^ P-- mu. 

creases with the time requ^ed to 'stot *T *' "'''°" 
reduction of speed, and these toLfTT''' "'*^ 
each other, yet mJess the .tn *' *^''^ *° "^utn 

friction duetto rSctionV^d ]Zu7' '""^ ''^"^"^ 

^peea is much greater than 



j_ 



ROLLING-STOCK. 4Q3 

5e due to time, and therefore the brake pressure must 
greater than the weight on the wheel, unless momentarily 
he speed is still ver}^ high. 

The adhesion betw^een wheels and rails varies from .20 
and over when the rail is dry. When wet and slippery 
fall to .18 or even .15. The use of sand will always 
above .20, and on a dry rail, when the sand is not blown 
y wind, it may raise it to .35 or even .40. 
Experiments were made w^ith an automatic valve by 
the brake-shoe pressure against the wheel should be 
I as the friction increased, but since (1) the essential 

L ment is that the friction produced by the brake-shoes 
ot exceed the adhesion between rail and wheel, and 
l) the rail-wheel adhesion is a very variable quantity, 
ing on whether the rail is wet or dry, it has been found 
icable to use such a valve, and that the best plan is to 

i to the engineer to vary the pressure, if necessary, by the 
:he brake- valve. 

MECHANISM OF BRAKES. 

Hand-brakes. The old style of brakes consists of brake- 
some type which are pressed against the wheel-treads 
ins of a brake-beam, which is operated by means of a 
imdlass and chain operating a set of levers. It is desir- 
at brakes shall not be set so tightly that the wheels 
3 locked, and then slide over the track, producing 
ces on them, which are very destructive to the 
tock and track afterward, on account of the impact 
ed at each revolution. With air-brakes the maxinmm 
of the brake-shoes can be quite carefully regulated, 
y are so designed that the maximum pressure exerted 

rlpair of brake-shoes on the wheels of any axle shall not 
I a certain per cent, of the weight carried by that axle 
\e car is empty, 90% being the figure usually adopted 
senger-cars and 70% for freight-cars. Consider the 
. freight-car of 100000 lbs. cap|pity, weighing 33100 lbs., 
tlbs. on an axle,- and equipped with a hand-brake which 
the levers and brake-beams, which are sketched in 

1 '. The dead weight on an axle is 8275 lbs.; 70% of 



404 



RAILROAD CONSTRUCTION. 



§ 335. 



this is 5792 lbs., which is the maximum allowable pressure 
per brake-beam, or 2896 lbs. per brake-shoe. With the dimen- 
sions shown, such a pressure will be produced by a pull of about 
1158 lbs. on the brake-chain. The power gained by the brake- 
wheel is not equal to the ratio of the brake-wheel diameter 
to the diameter of the shaft, about which the brake-chain 
winds, which is about 16 to IJ. The ratio of the circumfer- 
ence of the brake-wheel to the length of chain wound up by 
one complete turn would be a closer figure. The loss of efh- 




5792 



Fig. 205. — Sketch of Mechanism of Hand-brake. 



ciency in such a clumsy mechanism also reduces the effective 
ratio. Assuming the effective ratio as 6:1 it would require a 
pull of 193 lbs. at the circumference of the brake-wheel to 
exert 1158 lbs. pull on the brake-chain, or 5792 lbs. pressure 
on the wheels at B, and even this will not lock the wheels when 
the car is empty, much less when it is loaded. Note that the 
pressures at A and B are unequal. This is somewhat objec- 
tionable, but it is unavoidable with this simple form of brake- 
beam. More complicated forms to avoid this are sometimes 
used. Hand-brakes are, of course, cheapest in first cost, and 
even with the best of automatic brakes, additional mechanism 
to operate the brakes by hand in an emergency is always pro- 
vided, but their slow oj^ration when a quick stop is desired 
makes it exceedingly dangerous to attempt to run a train at 
high speed unless some automatic brake directly under the 
control of the engineer is at hand. The great increase in the 



§ 337. ROLLING-STOCK. 405 

average velocity of trains during recent years has only been 
rendered possible by the invention of automatic brakes. 

336. "Straight** air-brakes. The essential constructive fea- 
tures of this form of brake are (1) an air-pump on the engine, 
operated by steam, which compresses air into a reservoir on 
the engine; (2) a ''brake-pipe" running from the reservoir 
to the rear of the engine and pipes running under each car, 
the pipes having flexible connections at the ends of the cars 
and engine; (3) a cylinder and piston under each car which 
operates the brakes by a system of levers, the cylinder being 
connected to the brake-pipe. The reservoir on the engine 
holds compressed air at about 45 lbs. pressure. To operate the 
brakes, a valve on the engine is opened which allows the com- 
pressed air to flow from the reservoir through the brake-pipe 
to each cylinder, moving the piston, which thereby moves the 
levers and applies the brakes. The defects of this system are 
many: (1) With a long train, considerable time is required for 
the air to flow from the reservoir on the engine to the rear cars, 
and for an emergency-stop even this delay would often be 
fatal; (2) if the train breaks in two, the rear portion is not 
provided with power for operating the brakes, and a dangerous 
collision would often be the result; (3) if an air-pipe coupling 
bursts under any car, the whole system becomes absolutely 
helpless, and as such a thing might happen during some emer- 
gency, the accident would then be especially fatal. 

This form of brake has almost, if not entirely, passed out of 
use. It is here briefly described in order to show the logical 
development of the form which is now in almost universal use, 
the automatic. 

337. Automatic air-brakes. The above defects have been 
overcome by a method which may be briefly stated as follows: 
A reservoir for compressed air is placed under each car and the 
tender; whenever the pressure in these reservoirs is reduced 
for any reason, it is automatically replenished from the main 
reservoir on the engine; whenever the pressure in the brake- 
pipe is reduced for any cause (opening a valve at any point of 
its length, parting of the train, or bursting of a pipe or coupler), 
valves are automatically moved under each car to operate the 
piston and put on the brakes. All the brakes on the train are 
thus applied almost simultaneously. If the train breaks in two, 
both sections will at once have all the brakes applied automati- 



406 RAILROAD CONSTRUCTION. § 337. 

cally; if a coupling or pipe bursts, the brakes are at once applied 
and attention is thereby attracted to the defect; if an emer- 
gency should arise, such that the conductor desires to stop 
the train instantly without even taking time to signal to the 
engineer, he can do so by opening a valve placed on each car, 
which admits air to the train-pipe, which will set the brakes 
on the whole train, and the engineer, being able to discover 
instantly what had occurred, would shut off steam and do 
whatever else was necessary to stop the train as quickly as pos- 
sible. The most important and essential detail of this system 
is the ^'automatic triple valve" placed under each car. Quot- 
ing from the Westinghouse Air-brake Company's Instruction 
Book, ^'A moderate reduction of air pressure in the train-pipe 
causes the greater pressure remaining stored in the auxiliary 
reservoir to force the piston of the triple valve and its slide- 
valve to a position which will allow the air in the auxiliary 
reservoir to pass directly into the brake-cylinder and apply the 
brake. A sudden or violent reduction of the air in the train- 
pipe produces the same effect, and in addition causes supple- 
mental valves in the triple valve to be opened, permitting the 
pressure from the train-pipe to also enter the brake-cylinder, 
augmenting the pressure derived from the auxiliary reservoir 
about 20%, producing practically instantaneous action of the 
brakes to their highest efficienc}^ throughout the entire train. 
When the pressure in the brake-pipe is again restored to an 
amount in excess of that remaining in the auxiliary reservoir, 
the piston- and slide-valves are forced in the opposite direction 
to their normal position, opening communication from the train- 
pipe to the auxiliary reservoir, and permitting the air in the 
brake-cyhnder to escape to the atmosphere, thus releasing the 
brakes. If the engineer wishes to apply the brake, he moves 
the handle of the engineer's brake-valve to the right, which 
first closes a port, retaining the pressure in the main reservoir, 
and then permits a portion of the air in the train-pipe to escape. 
To release the brakes, he moves the handle to the extreme 
left, which allows the air in the main reservoir to flow freely 
into the brake-pipe, restoring the pressure therein." 

338. Tests to measure the efficiency of brakes. Let v repre- 
sent the velocity of a train in feet per second: TT^, its weight; 
F, the retarding force due to the biakes; c?, the distance in feet 
required to make a stop; and g, the acceleration of gravity 



§ 339. ROLLING-STOCK. 407 

(32.16 feet per square second); then the kinetic energy pos- 
sessed by the train (disregarding for the present the rotative 

kinetic energy of the wheels) = -^ — . The work done in stop- 

ping the tvsiin =Fd. .'. Fd = ~^. The ratio of the retarding 

force to the weight, ' 

F v^ v^ 

In order to compare tests made under varying conditions, the 
ratio F ^W should be corrected for the effect of grade ( + or — ), 
if any, and also for the proportion of the weight of the train 
which is on braked wheels. For example, a train weighed 
146076 lbs., the proportion on braked wheels was 67%, speed 
60 feet per second, length of stop 450 feet, track level. Sub- 
stituting these values in the above formula, we find {F ^W) 
= .124. This value is really unduly favorable, since the ordi- 
nary track resistance helps to stop the train. This has a value 
of from 6 to 20 lbs. per ton, averaging say 10 lbs. per ton dur- 
ing the stop, or .005 of the weight. Since the effect of this is 
srnall and is nearly constant for all trains, it may be ignored 
in comparative tests. The grade in this case was level, and 
therefore grade had no effect. But since only 67% of the 
weight was on braked wheels, the ratio, on the basis of all the 
wheels braked, or of the weight reduced to that actually on the 
braked wheels, is 0.124 --.67 =0.185. This was called a ''good" 
stop, although as high a ratio as 0.200 has been obtained. 

339. Brake-shoes. Brake-shoes were formerly made of 
wrought iron, but when it was discovered that cast-iron shoes 
would answer the purpose, the use of wrought-iron shoes was 
abandoned, since the cast-iron shoes are so much cheaper. A 
cheap practice is to form the brake-shoe and its head in one 
piece, which is cheaper in first cost, but when the wearing-sur- 
face is too far gone for further use, the whole casting must be 
renewed. The ''Christie" shoe, adopted by the Master Car 
Builders' Association as standard, has a separate shoe which 
is fastened to the head by means of a wrought-iron key. The 
shoe is beveled \" in a width of 3f" to fit the coned wheel. 
This is a greater bevel than the standard coning of a car-wheel. 
It is perhaps done to allow for some bending of the brake- 



408 RAILROAD CONSTRUCTION. § 339. 

beam and also so that the maximum pressure (and wear) should 
come on the outside of the tread, rather than next to the flange, 
where it might tend to produce sharp flanges. By concen- 
trating the brake-shoe wear on the outer side of the tread, the 
wear on the tread is more nearly equalized, since the rail wears 
the wheel-tread chiefly near the flange. This same idea is 
developed still further in the '^ flange-shoes,'^ which have a 
curved form to fit the wheel-flange and which bear on the 
wheel on the flange and on the outside of the tread. It is 
claimed that by this means the standard form of the tread is 
better preserved than when the wear is entirely on the tread. 
The Congdon brake-shoe is one of a type in which wrought- 
iron pieces are inserted in the face of a cast-iron shoe. It is 
claimed that these increase the life of the shoe. 



CHAPTER XVI. 

TRAIN RESISTANCE. 

340. Classification of the various forms. The various resist- 
ances which must be overcome by the power of the locomotive 
may be classified as follows : 

(a) Resistances internal to the locomotive, which include fric- 
tion of the valve-gear, piston- and connecting-rods, journal 
friction of the drivers; also all the loss due to radiation, con- 
densation, friction of the steam in the passages, etc. In short, 
these resistances are the sum-total of the losses by which the 
power at the circumference of the drivers is less than the power 
developed by the boiler. 

(b) Velocity resistances, which include the atmospheric resist- 
ances on the ends and sides; oscillation and concussion resist- 
ances, due to uneven track, etc. 

(c) Wheel resistances, which include the rolling friction be- 
tween the wheels and the rails of all the wheels (including the 
drivers) ; also the journal friction of all the axles, except those 
of the drivers. 

(d) Grade and curve resistances, which include those resist- 
ances which are due to grade and to curves, and which are not 
found on a straight and level track. 

(e) Brake resistances. As shown later, brakes consume 
power and to the extent of their use increase the energy to 
be developed by the locomotive. 

(/) Inertia resistances. The resistance due to inertia is not 
generally considered as a train resistance because the energy 
which is stored up in the train as kinetic energy may be util- 
ized in overcoming future resistances. But in a discussion 
of the demands on the tractive power of the engine, one of the 
chief items is the energy required to rapidly give to a starting 
train its normal velocity. This is especially true of suburban 
trains, which must acquire speed very quickly in order that 

409 



410 RAILROAD CONSTRUCTION. § 341. 

their general average speed between termini ma}- be even reason- 
ably fast. 

341. Resistances internal to the locomotive. These are re- 
sistances which do not tax the adhesion of the drivers to the 
rails, and hence are frequently considered as not being a part 
of the train resistance properly so called. If the engine were 
considered as lifted from the rails and made to drive a belt 
placed around the drivers, then all the power that reached the 
belt would be the power that is ordinarily available for adhe- 
sion, while the remainder would be that consumed internally 
by the engine. The power developed b}^ an engine may be 
obtained by taking indicator diagrams which show the actual 
steam pressure in a cylinder at any part of a stroke. From 
such a diagram the average steam pressure is easily obtained, 
and this average pressure, multiplied by the length of the stroke 
and by the net area of the piston, gives the energy developed 
by one half-stroke of one piston. Four times this product 
divided b}^ 550 times the time in seconds required for one stroke 
gives the "indicated horse-power" Even this calculation 
gives merely the power behind the piston, which is several per 
cent, greater than the power which reaches the circumference 
of the drivers, owing to the friction of the piston, piston-rod, 
cross-head, connecting-rod bearings, and driving-wheel jour- 
nals. (See § 322, Chapter XV.) By measuring the amount 
of water used and turned into steam, and b}^ noting the boiler 
pressure, the energy possessed by the steam used is readily 
computed. The indicator diagrams w^ill show the amount of 
steam that has been effective in producing power at the cylin- 
ders. The steam accounted for by the diagrams will ordinarily 
amount to 80 or 85% of the steam developed by the boiler, 
and the other 15 or 20% represents the loss of energy due to 
radiation, condensation, etc. From actual tests it has been 
found that the power consumed by an engine running light is 
about 11%. of that required by the engine when working hard 
in express freight service. But since the engine resistances 
(friction, etc.) are increased when it is pulling a load, it was 
estimated, after allowing for this fact, that about 15 or 16% 
of the power developed by the pistons was consumed by the 
engine, leaving about 84 to 85% for the train. 

342. Velocity resistances, (a) Atmospheric. This consists of 
the head and tail resistances and the side resistance. The hep-d 



§ 343. TRAIN RESISTANCE. 411 

and tail resistances are neaily constant for all trains of given 
velocity, varying but slightly with the varying cross-sections 
of engines and cars. The side resistance varies with the length 
of the train and the character of the cars, box-cars or flats, etc. 
Vestibuling cars has a considerable effect in reducing this side 
resistance by preventing much of the eddying of air-currents 
between the cars, although this is one of the least of the ad 
vantages of vestibuling. Atmospheric resistance is generally 
assumed to vary as the square of the velocity, and although 
this may be nearly true, it has been experimentally demon 
strated to be at least inaccurate. The head resistance is gen 
erally assumed to vary as the area of the cross-section, but this 
has been definitely demonstrated to be very far from true. A 
freight-train composed partly of flat-cars and partly of box- 
cars will encounter considerably more atmospheric resistance 
than one made exclusively of either kind, other things being 
equal. The definite information on this subject is very unsat- 
isfactory, but this is possibly due to the fact that it is of little 
practical importance to know just how much such resistance 
amounts to. 

(6) Oscillatory and concussive. These resistances are con- 
sidered to vary as the square of the velocity. Probably this 
is nearly, if not quite, correct on the general principle that such 
resistances are a succession of impacts and the force of impacts 
varies as the square of the velocity. These impacts are due to 
the defects of the track, and even though it w^ere possible to 
make a precise determination of the amount of this resistance 
in any particular case, the value obtained w^ould only be true 
for that particular piece of track and for the particular degree 
of excellence or defect which the track then possessed. The 
general improvement of track maintenance during late years 
has had a large influence in increasing the possible train-load 
by decreasing the train resistance. The expenditure of money 
to improve track will give a road a large advantage over a 
competing road with a poorer track, by reducing train resist- 
ance, and thus reducing the cost of handling traffic. 

343. Wheel resistances, (a) Rolling friction of the wheels. 
To determine experimentally the rolling friction of wheels, 
apart from all journal friction, is a very difficult matter and 
has never been satisfactorily accomplished. Theory as well 
as practice shows that the higher and the more perfect the 



412 RAILROAD CONSTRUCTION. § 343. 

elasticity of the wheel and the surface, the less will be the roll- 
ing friction. But the determination, if made, would be of 
theoretical interest only. 

The combined effect of rolling friction and journal friction 
is determinable with comparative ease. From the nature of 
the case no great reduction of the rolling friction by any device 
is possible. It is only a very insignificant part of the total 
train resistance. 

(6) Journal friction of the axles. This form of resistance has 
been studied quite extensively by means of the measurement 
of the force required to turn an axle in its bearings under 
various conditions of pressure, speed, extent of lubrication, 
and temperature. The following laws have been fairly well 
established: (1) The coefficient of friction increases as the pres- 
sure diminishes; (2) it is higher at very slow speeds, gradually 
diminishing to a minimum at a speed corresponding to a train 
velocity of about 10 miles per hour, then slowly increasing 
with the speed; it is ver^^ dependent on the perfection of the 
lubrication, it being reduced to one sixth or one tenth, when the 
axle is lubricated by a bath of oil rather than by a mere pad 
or wad of waste on one side of the journal; (3) it is much lower 
at higher temperature, and vice versa. The practical effect of 
these laws is shown by the observed facts that (1) loaded cars 
have a less resistance per ton than unloaded cars, the figures 
being (for speeds of about 10 to 20 miles per hour) : 

For passenger- and loaded freight-cars. . . 4 lbs. per ton 

* ' empty freight-cars 6 " ' ' *' 

'' street-cars 10 '' '' '' 

'' freight-trucks without load 14 ^^ '' '' 

(2) When starting a train, the resistances are about 20 lbs. 
per ton, notwithstanding the fact that the velocity resistances 
are practically zero; at about 2 miles per hour it wall drop to 
10 lbs. per ton and above 10 miles per hour it may drop to 
4 lbs. per ton if the cars are in good condition. (3) The re- 
sistance could probably be materially lowered if some practicable 
form of journal-box could be devised which would give a more 
perfect lubrication. (4) It is observed that freight-train loads 
must be cut down in winter by about 10 or 15% of the loads 
that the same engine can haul over the same track in summer. 
This is due partly to the extra roughness and inelasticity of the 



§344. 



TRAIN RESISTANCE. 



413 



track in winter, and partly to increased radiation from the 
engine wasting some energy, but this will not account for all 
of the loss, and the effect, which is probably due largely to the 
lower temperature of the journal-boxes, is very marked and 
costly. It has been suggested that a jacketing of the journal- 
boxes, which w^ould prevent rapid radiation of heat and enable 
them to retain some of the heat developed by friction, would 
result in a saving amply repaying the cost of the device. 

Roller journals for cars have been frequently suggested, and 
experiments have been made with them. It is found that they 
are very effective at low velocities, greatly reducing the start- 
ing resistance, which is very high with the ordinary forms of 
journals. But the advantages disappear as the velocity in- 
creases. The advantages also decrease as the load is increased, 
so that with heavily loaded cars the gain is small. The excess 
of cost, for construction and maintenance has been found to be 
more than the gain from power saved. 

344. Grade resistance. The amount of this may be com- 
puted with mathematical exactness. Assume that the ball 
or cylinder (see Fig. 206) is being drawn up the plane. If W 




Fig. 206. 
is the weight, N the normal pressure against the rail, and G 
the force required to hold it or to draw it up the plane with 
uniform velocity, the rolling resistances being considered zero 
or considered as provided for by other forces, then 

G:W:h:d, or G = ^; 

but for all ordinary railroad grades, d=c to within a tenth of 

Wh * 

1%, i.e., G = = TF X rate of grade. In order that the student 

may appreciate the exact amount of this approximation the per- 
centage of slope distance to its horizontal projection is given in 
the following tabular form: 



414 



RAILROAD CONSTRUCTION. 



§344. 



Grade in per cent. 


1 


2 


3 


4 


5 


Slope di.st.^^OQ 

hor. dist. 


100.005 


100.020 


100.045 


100.080 


100.125 




Grade in per cent. 


6 


7 


8 


9 


10 


Slope dist. ^jQQ 

hor. dist. 


100.180 


100.245 


100.319 


100.404 


100.499 



This shows also the error on various grades of measuring with 
the tape on the ground rather than held horizontally. Since 
almost all railroad grades are less than 2% (where the error 
is but .02 of 1%), and anything in excess of 4% is unheard 
of for normal construction, the error in the approximation 
is generally too small for practical consideration. 

If the rate of grade is 1 : 100, G = WXji-^j i-^-, 6^ = 20 lbs. 
per ton ; .'. for any per cent, of grade, G = (20 X per cent, of grade) 
pounds per ton. When moving up a grade this force G is to 
be overcome in addition to all the other resistances. When 
moving doAvn a grade, the force G assists the motion and may 
be more than sufficient to move the train at its highest allow- 
able velocity. The force required to move a train on a level 
track at ordinary freight-train speeds (say 20 miles per hour) 
is about 7 lbs. per ton. A down grade of /^ of 1% will fur- 
nish the same power; therefore on a down grade of 0.35%, a 
freight-train would move indefinitely at about 20 miles per hour. 
If the grade were higher and the train were allowed to gain 
speed freely, the speed would increase until the resistance at 
that speed would equal W times the rate of grade, when the 
velocity w^ould become uniform and remain so as long as the 
conditions were constant. If this speed was higher than a 
safe permissible speed, brakes must be applied and power 
wasted. The fact that one terminal of a road is considerably 
higher than the other does not necessarily imply that the extra 
power needed to overcome the difference of elevation is a 
total waste of energy, especially if the maximum grades are 
so low that brakes will never need to be applied to reduce a 
dangerously high velocity, for although more power must be 



§ 347. TRAIN RESISTANCE. 415 

used in ascending the grades, there is a considerable saving of 
power in descending the grades. The amount of this sa\dng 
will be discussed more fully in Chapter XXIII. 

345. Curve resistance. Some of the principal laws will be 
here given without elaboration. A more detailed discussion 
will be given in Chapter XXII. 

(a) While the total curve resistance increases as the degree 
of curve increases, the resistance 'per degree of curve is much 
greater for easy curves than for sharp curves; e.g., the resist- 
ance on the excessively sharp curves (radius 90 feet) of the 
elevated roads of New York City is very much less per degree 
of curve than that oti curves of 1° to 5°. (h) Curve resistance 
increases with the velocity, (c) The total resistance on a 
curve depends on the central angle rather than on the radius; 
I.e., two" curves of the same central angle but of different radius 
would cause about the same total curve resistance. This is 
partly explained by the fact that the longitudinal slipping will 
be the same in each case. (See § 311, Cliapter XV.) In each 
case also the trucks must be twisted around and the wheels 
slipped laterally on the rails by the same amount J^. (See 
§ 312, Chapter XV.) 

346. Brake resistances. If a down grade is excessively steep 
so that brakes must be applied to prevent the train acquiring 
a dangerous velocity, the energy consumed is hopelessly lost 
without any compensation. WTien trains are required to make 
frequent stops and yet maintain a high average speed, consid- 
erable power is consumed by the application of brakes in stop- 
ping. All the energy which is thus turned into heat is hope- 
lessly lost, and in addition a very considerable amount of steam 
is draw^n from the boiler to operate the air-brakes, which con- 
sume the power already developed. It can be easih^ demonstrated 
that engines drawing trains in suburban service, making fre- 
quent stops, and yet developing high speed between stops, will 
consume a very large proportion of the total power developed 
by the use of brakes. Note the double loss. The brakes con- 
sume power already developed and stored in the train as kinetic 
or potential energy, while the operation of the brakes requires 
additional steam power from the engine. 

347. Inertia resistance. The two forms of train resistance 
which under some circumstances are the greatest resistances 
to be overcome by the engine are the grade and inertia resist- 



416 RAILROAD CONSTRUCTION. § 347. 

ances, and fortunately both of these resistances may be com- 
puted with mathematical precision. The problem may be 
stated as follows: What constant force P (in addition to the 
forces required to overcome the various frictional resistances, 
etc.) will be required to impart to a body a velocity of v feet 
per second in a distance of s feet? The required number of 
foot-pounds of energy is evidently Ps. But this work imparts 

a kinetic energy which may be expressed by -^—^ Equating 

^9 

these values, we have Ps = -^r—, or 

2g 

^ = ^ <138) 

The force required to increase the velocity from v^ to V2 may 

W 
likewise be stated as P=;^ — (^^^—^i')- Substituting in the 

formula the values TF=2000 lbs. (one ton), ^=32.16, and s = 
5280 feet (one mile), we have 

P = .00588(V-V). 
Multiplying by (5280-^3600)2 to change the unit of velocity 
to miles per hour, we have 

P = .01267(72^ -V,^), 

But this formula must be modified on account of the rotative 
kinetic energy which must be imparted to the wheels of the cars. 
The precise additional percentage depends on the particular 
design of the cars and their loading and also on the design of 
the locomotive. Consider as an example a box-car, 60000 lbs. 
capacity, weighing 33000 lbs. The wheels have a diameter 
of 36" and their radius of gyration is about 13'''. Each wheel 
weighs 700 lbs. The rotative kinetic energy of each wheel is 
4877 ft.-lbs. when the velocity is 20 miles per hour, and for 
the eight wheels it is 39016 ft.-lbs. For greater precision 
(really needless) we may add 192 ft.-lbs. as the rotative kinetic 
energy of the axles. When the car is fully loaded (weight 
93000 lbs.) the kinetic energy of translation is 1,244,340 ft.-lbs.; 
when empty (weight 33000 lbs.) the energy is 441540 ft.-lbs. 
The rotative kinetic energy thus adds (for this particular 
car) 3.15% (when the car is loaded) and 8.9% (when the car 
is empty) to the kinetic energy of translation. The kinetic 



§ 347. TRAIN RESISTANCE. 417 

energy which is similariy added, owing to the rotation of the 
wheels and axles of the locomotive, might be similarly com- 
puted. For one type of locomotive it has been figured at about 
8%. The variations in design, and particularly the fluctua- 
tions of loading, render useless any great precision in these 
computations. For a train of ^^ empties" the figure would be 
high, probably 8 to 9%; for a fully loaded train it will not 
much exceed 3%. Wellington considered that 6% is a good 
average value to use (actually used 6.14% for ^'ease of compu- 
tation"), but considering (a) the increasing proportion of live 
load to dead load in modern car design, (h) the greater care 
now used to make up full train-loads, and (c) the fact that 
full train-loads are the critical loads, it would appear that 5% 
is a better average for the conditions of modern practice. Even 
this figure allows something for the higher percentage for the 
locomotive and something for a few empties in the train. There- 
fore, adding 5% to the coefficient in the above equation, we 
have the true equation 

P = .0133(TV-1^V), (139) 

in which V2 and Fj are the higher and lower velocities respec- 
tively in miles per houVj and P is the force required per ton to 
impart that difference of velocity in a distance of one mile 
If more convenient, the formula may be used thus: 

P.=^^(TV-F,=), .... (140) 

in which 5 is the distance in feet and Pi is the corresponding 
force. 

As a numerical illustration, the force required per ton to 
impait a kinetic energy due to a velocity of 20 miles per hour 
in a distance of 1000 feet will equal 

_ 70.224(400-0) _ 
P: 1000 ^^^^^■' 

which is the equivalent (see § 344) of a 1.4% grade. Since the 
velocity enters the formula as V^, w'hile the distance enters only 
in the first power, it follows that it will require four times the 
force to produce twn'ce the velocity in the same distance, or 
that with the same force it will require four times the distance 
to attain twice the velocity. 



418 RAILROAD CONSTRUCTION. § 347. 

As another numerical illustration, if a train is to increase its 
speed from 15 miles per hour to 60 miles per hour in a distance 
of 2000 feet, the force required (in addition to all the other 
resistances) will be 

70.224(3600-225) ^^^3 ^Q ^^^ ,^^^ 
' 2000 ^ 

This is equivalent to a 5.9% grade and shows at once that it 
would be impossible unless there w^ere a very heavy down 
grade, or that the train was very light and the engine very 
powerful. 

348. Dynamometer tests. These are made by putting a 
''dynamometer-car" between the engine and the cars to be 
tested. Suitable mechanism makes an automatic record of 
the force which is transmitted through the dynamometer at 
any instant, and also a record of the velocity at any instant. 
One of the practical difficulties is the accurate determination 
of the velocity at any instant when the velocity is fluctuating. 
AVhen the velocity is decreasing, the kinetic energy of the train 
is being turned into work and the force transmitted through the 
dynamometer is less than the amount of the resistance which 
is actually being overcome. On the other hand, when the 
velocity is increasing, the dynamometer indicates a larger 
force than that required to overcome the resistances, but the 
excess force is being stored up in the train as kinetic energy. 
Grade has a similar effect, and the force indicated by the dy- 
namometer may be greater or less than that required at the 
given velocity on a level by the force which is derived from, 
or is turned into, potential energy. Therefore the resistance 
indicated by the dynamometer of a train will not be that on a 
level track at uniform velocity, unless the track is actually 
level and the velocity really uniform. 

Dynamometer tests under other circumstances are there- 
fore of no value unless it is possible to determine the true 
velocity at any instant and its rate of change, and also to de- 
termine the grade. Of course, the grade is easily found. An 




20 ;30 40 50 

Velocity in miles per hour 

{To face page 4 IS.) 



GO 



§ 350. 



TRAIN RESISTANCE. 



419 



allowance for an increase or decrease of kinetic or potential 
energy must therefore be made before it is possible to know 
how much force is being spent on the ordinary resistances. 

349. Gravity or " drop " tests. Dynamometer tests require 
the use of a dynamometer which is capable of measuring a 
force of several thousands of pounds, and w^hich therefore 
cannot determine such values with a close percentage of accu- 
T^cy, especially if the force is small. A drop test utilizes the 
force of gravity w^hich may be measured wdth mathematical 
accuracy. The general method is to select a stretch of track 
Avhich has a uniform grade of about 0.7% and which is prefer- 
ably straight for two or three miles. On such a grade cars 
wdth running gear in good condition may be started by a push. 
The velocity will gradually increase until at some velocity, 
depending on the resistances encountered, the cars w^ill move 
uniformly. The only work requiring extreme care with this 
method is the determination of the velocity. If the velocity 
is fluctuating, as it is during the time when it is of the greatest 
importance to know the velocit}', it is not sufficient to deter- 
mine the time required to run some long measured distance, 
for the average velocity thus obtained would probably differ 




Fig. 207. — Loss in Velocity-head. 

considerably from the velocity at the beginning and end of that 
space. If the train consists of five cars or more, the velocity 
may be determined electrically (as described by Wellington 
in his "Economic Location,'' etc., p. 793 et seq.) from the 
automatic record made on a chronograph of the passage of the 
first w^heel and the last, the chronograph also recording auto- 



420 RAILROAD CONSTRUCTION § 350. 

matically the ticks of a clock beating seconds. From this the 
exact time of the passage of the first and last wheels of the 
train of cars may be determined to the tenth or twentieth of a 
second. 

Velocity -head. From theoretical mechanics we know^ that 
if a body descends through any path by the action of gravity, 
and is unaffected by friction, its velocity at any point in the 
direction of the path of motion is V^\/2gh. If the body is 
retarded by resistances, its velocity at any point will be less 
than this. If AM, Fig. 207, represents any grade (exaggerated 
of course), then BJ, CK, etc., represent the actual fall at any 

point. Let BF represent the fall h^y determined from h^ = j~y 

in which v^ is the actual observed velocity at J. Then /F = the 
velocity-head consumed by the resistances between A and ./. 
If the train continues to /v, the corresponding A, is CG) the 
remaining fall GK consists of GN (=JF, which is the velocity- 
head lost back of J) and NK, the velocity-head lost between J 
and K. At some velocity (Vn) on any grade, the velocity 
will not further increase and the line AFGHI will then be hori- 
zontal and at a distance (hn)=EI below A . . , E. The grade 
AM is the ^' grade of repose" for that velocity (Fn); i.e., it is 
the grade that w^ould just permit the train to move indefinitely 
at the velocity Vn. The broken line AFGHI should really be 
a curve, and the grade of repose at any point is the angle between 
AM and the tangent to that curve at the given point. The 
''grade of repose" by its definition gives the total resistance 
of the train at the particular velocity, or multiplying the grade 
of repose in per cent, by 20 gives the pounds per ton of resist- 
ance. Thus being able to determine the total resistance in 
pounds per ton at any velocity, the variation of total resistance 
with velocity may be determined, and then by varying the 
resistances, using different kinds of cars, empty and loaded, 
box-cars and flats, the resistances of the different kinds at 
various velocities may be determined. 

350. Formulae for train resistance. These are generally given 
in one of the forms 

R = aV + c, ... (1)1 

R = bV'-hc, ... (2) I ... . (141) 

R=^aV + bV^-^c, . . (3) 



§ 350. TRAIN RESISTANCE. 421 

in which R is the resistance in pounds per ton, a and h are coeffi- 
cients to be determined, V is the velocity in miles per hour, and 
c is a constant, also to be determined. These formulae disregard 
grade and curve resistances, inertia resistance and the active 
resistance (or assistance) of windy as distinct from mere atmos- 
pheric resistance. In short, they are supposed to give the re- 
sistance of a train moving at a uniform velocity over a straight 
and level track, there being no appreciable wind. 

The various formulae are sometimes based directly on experi- 
ments made by the proposer of the formula ; sometimes they are 
deduced from a mere study of the results of one or more series 
of tests made by others. Unfortunately for either method, no 
one investigator has ever been able to make tests which are so 
thorough and made under such a w^ide range of conditions that 
his results may be considered as conclusive, while a student of 
the tests of others is handicapped by a lack of knowledge of 
precise conditions, which, if fully understood, would perhaps 
permit some reconciliation of the very discordant figures which 
are reported. As already intimated, the condition of the 
rolling stock, the unit weight on the axles, the lubrication of the 
axles, the length of the train in relation to its weight and the 
condition of the track, which involves the weight of rail, spacing 
and size of ties, tamping of ties, etc., all have their influence in 
modifying the apparent resistance. There is also good reason to 
believe that the effect of grade, curvature, and changing velocity 
has not been properly allowed for in deducing many of the 
formulae. In view of all these considerations, it may be con- 
sidered as demonstrated that no one formula, and especially a 
simple formula, will represent the resistance for all conditions. 
But, since some of the calculations of railroad economics are 
absolutely dependent on the law of tractive resistance, some 
law must be deduced with sufficient accuracy for the purpose. 
Fortunately several of the formulae are amply accurate for such 
purposes. A report of a committee of the A. R. E. & M. W. 
Assoc. (1907) quoted sixty-one different formulae which have been 
suggested. Some of these are chiefly of historical value, since 
they were deduced from tests made many years ago with track 
and rolling stock very dissimilar from those in use at the present 
time. Such formulae will therefore be omitted. For con- 
venience of comparison, all formulae will be changed (if neces- 
sary) from the original statement of them so that they give the 



422 



RAILROAD CONSTRUCTION. 



§ 350. 



resistance per ton of 2000 pounds. The coefficients of V and 
V^ will be given decimally. Other notation occasionally used 
is as follows: 

^ = weight of train in tons of 2000 pounds; 
L == length of train in feet ; 
n = number of cars in train; 
A = area of front of train in square feet. 

(a) Formulae of the first class: R = aV + c. Among those 
most commonly used are the following: 



Engineering News, 
Baldwin locomotive, 
New York Central, 
Henderson, 



R = 0.25V + 2.0 (142) 

i? = 0.17F + 3.0 (143) 

7? = 0.11F+1.8, (144) 

72 = 0.257 + ^-+ 0.5 (145) 



Although Henderson ^s formula is in a class by itself, on account 
of the extra term, and although it is not applicable to general 
use, when the character of the trains cannot be estimated, it 
is perhaps more accurate than the others. It is apparently not 
intended for use at very low velocities. 

(6) FormulsD of the second class: R^bV^-hc: 



Crawford, 
WolfP, 
Henderson, 
Forney, 



R = 0.00214^^2.5 (146) 

7^ = 0.00357724-2.7 (147) 

7^ = 0.004617^ + 3.0 (148) 

7? = 0.0058572 + 4.0 (149) 



Wel- 
ling- 
ton 



57T72 

72 = 0.005672 + ^—— +3.9 (for loaded flat cars) 

6472 
7? = 0.007572 + —^ + 3.9 (for loaded boxcars) 

571/2 
72 = 0.008372 + ^—— +6.0 (for empty flat cars) 

6472 
72 = 0.010672 + ^—— +6.0 (for empty box cars) 



u (150) 



Notice in formulae (150) the additioMal jourj>al resistance 
(indicated by the constant term) for unloaded cars. The second 



§ 350. TRAIN RESISTANCE. 423 

term evidently indicates the atmospheric resistance. The first 
term allows for the oscillatory resistances. Assuming the con- 
stant term and the coefficients to have been correctly deter- 
mined, these formulae should be better than the others, since 
a choice of formulae can be made depending on the conditions. 
A train consisting partly of box-cars and partly of flat-cars 
will have a higher resistance than is shown by any of the above 
formulae (and not a mean value), on account of the increased 
atmospheric resistance acting on the irregular form of the train, 
(c) Formulae of the third class: R = aV + bV^ + c: 

W.N.Smith, jg = ai77+ ^ ^ +3.0;. . . . (151) 
VonBorries, i^ = 0.04 7 + 0.001672-}- 3.0; .... (152) 
Lundie, i2 = 0.247+^^+4.0; (153) 

Sprague, 7^ = 0.177+ ^^ +4.0 (154) 



Although several formulae have been proposed which involve 
the area of the front of the train in order to allow more definitely 
for the atmospheric resistance, only one of these (151) has been 
quoted. In applying this formula, the proper value to choose 
for A is somewhat indefinite, since the shape of the front of the 
train will make a considerable difference in the atmospheric 
resistance encountered. The area will vary from 80 to 100 
square feet. In the comparison of the formulae given below, 
A will be assumed as 100 square feet. In order to compare 
these resistances, the values of R for the various speeds of 10, 
20, 30, 40, 50, and 60 miles per hour will be computed by 
these formulae on the basis of a train of twelve cars, having a 
length of 480 feet, and a weight of 600 tons. Therefore in 
applying the formula, ^ = 600, i: = 480, n = 12, and A = 100. In 
order to apply formula (150) to this case, it will be assumed 
that this train consists of loaded box-cars, and therefore we 
must apply the second of that group of formulae. Computing 
the resistance according to these several formulae^ we may 
tabulate the results as given below: 



424 



RAILROAD CONSTRUCTION. 



350. 



Formula. 


Velocity in miles per hour. 




10 


20 


30 


40 


50 


60 


142 
143 
144 
145 

146 
147 
1 148 
149 
150 

151 
152 
153 
154 


2700 
2800 
1747 
2400 

1628 
1834 
2077 
2751 
2854 

2845 
2136 
4320 
3453 


4200 
3800 
2413 
3900 

2014 
2477 
2906 
3804 
4396 

3940 
2664 
7200 
4573 


5700 
4800 
3080 
5400 

2656 
3548 
4289 
5559 
6966 

5085 

3384 

11040 

5760 


7200 
5800 
3747 
6900 

3554 
5047 
6226 
8116 
10564 

6280 

4296 

15840 

7013 


8700 
6800 
4413 
8400 

4710 
6975 

8715 
11175 
15188 

7525 

5400 

19440 

8333 


10200 
7800 
5080 
9900 

6122 

9331 

11746 

15036 

20844 

8820 

6696 

28080 

9720 



Although there is a fair agreement among the results for 
ordinary velocities^ it should be said, in fairness to the proposers 
of the various formulae, that some of them evidently were not 
designed for use at high velocities such as 60 miles per hour. 

Another method of comparing formulae is to plot them on 
cross-section paper, using velocities as abscissae and resistances 
as ordinates. For general use this method may only be applied 
to formulae which do not involve the weight, length or area of 
the train nor the number of cars. All of the above formulae 
have thus been plotted on Plate IX, with the exception of Nos. 
145, 150, 151, 153, and 154. 



§ 350. 



TRAIN RESISTANCE. 



425 



25^ 
































PLA 


TE 


X. 


TRA 


IN F 


:ESI{ 


3TAI 


MCE 




















































































/ 






—20 






















/ 


























/ 


1 








^ 


















ST 


/ 




























k 
























/?' 


/ 






1-15- 




















'^ 


// 


<$> . 






















J 


1/ 






a 
















/ 




/ 


/ 


r 




















L 


4 


/ 


A 


<& 




















'4 


^/r 




y 






1-10 

1 














/ 


/ ^ 




/' 




^-, 














A 


/ 


A^ 


[cP / 




/ 


7 
















// 


^ 




# 




/ 
















/; 


^. 


/ 


/ 








v^ 












/ y 


^ 


/ 


A 






<> 








sr 






J 


y 


/ 


^ 


r> 


^^ 


r 
















/ 


^ 


^ 


^A 


^*- 


\^ 


















^ 


^ 


^^ 










































































































1 


) 




) 3p 
loclty in miles \ 


4p 
er hour 


50 


60 



CHAPTER XVII 

COST OF RAILROADS. 

351. General considerations. Although there are many ele- 
ments in the cost of railroads which are roughly constant per 
mile of road, yet the published reports of the cost of railroads 
differ very widely. The variation in the figures is due to several 
causes, (a) Economy requires that a road shall be operated 
and placed on an earning basis as soon as possible. Therefore 
the reported cost of a road during the first few years of its 
existence is somewhat less than that reported later. This is 
well illustrated when a long series of consecutive reports from 
an old-established road is available; nearly every year there 
will be shown an addition to the previous figures. And this 
is as it should be. The magnificent road-beds of some old 
roads cannot be the creation of a single season. It takes many 
years to produce such settled perfect structures, (b) A large 
part of the variation is due to a neglect to charge up " permanent 
improvements'' as additions to the cost of the road. For the 
first few years of the life of a road a great deal of work is done 
which is in reality a completion of the work of construction, 
and yet the cost of it is buried under the item ^'maintenance 
of way." For example, a long wooden trestle is replaced by 
an earth embankment and a culvert. Since the original trestle 
is to be considered a temporary structiu-e, the excess of the 
cost of the permanent structure over that of the temporary 
structure should evidently be considered as an addition to the 
cost of the road. But if the filling-in was done slowh^, a few 
train-loads at a time, and the work scattered over many years, 
the cost of operating the "mud-train'' has perhaps been buried 
under "maintenance" charges, (c) The reports from which 
many of the following figures were taken have not always 
analyzed the items of cost with the same detail as has been 
here attempted, and to that is probably due many of the varia- 
tions and apparent discrepancies. 

426 



§ 352 COST OF RAILROADS. 427 

The various items of cost will be classified as follows: 

1. Preliminary financiering. 

2. Surveys and engineering expenses. 

3. Land and land damages. 

4. Clearing and grubbing. 

5. Earthwork. 

6. Bridges, trestles, and culverts 

7. Trackwork. 

8. Buildings and miscellaneous structures. 

9. Interest on construction. 
10. Telegraph line. 

352. Item I. PRELIMINARY FINANCIERING. The COSt of this 
preliminary work is exceedingly variable. The work includes 
the clerical and legal work of organization, printing, engraAdng 
of stocks and bonds, and (sometimes the most expensive of all) 
the securing of a charter. This sometimes requires special 
legislative enactments, or may sometimes be secured from a 
State railroad commission. It has been estimated that about 
2% of the railway capital of Great Britain has been spent in 
Parliamentary expenses over the charters. These expenses 
are usuall}^ but a small percentage of the total cost of the enter- 
prise, but for important lines the gross cost is large, while the 
amount of money thus spent by organizations which have 
never succeeded in constructing their roads is, in the aggregate, 
an enormous amount, although it is of course not ascertainable 
by any investigator. 

Another occasional feature of the financing of a road must be 
kept in mind. The promoters of a railroad enterprise frequently 
endeavor to limit their own personal expenditures to the purely 
preliminary expenses as mentioned above. The project, after 
having been surveyed, mapped, and written up in a glowing 
''prospectus," is submitted to capitalists, in the endeavor to 
have them furnish money for construction, the money to be 
secured by bonds. If the project will stand it, the amount of 
the bond issue is made sufficient to pay the entire cost of the 
road, even with a discount of perhaps 15%. The bond issue 
may also provide for a very generous commission to the broker 
who is the intermediary between the promoters and the capi- 
talists. The bond issue may even provide for repaying the 
promoters for their preliminary expenses. Frequently a con- 
siderable proportion of the capital stock goes to the capitalists 



428 RAILROAD CONSTRUCTION. § 352. 

who take the bonds, the promoters retaining only such propor- 
tion as may be agreed upon. In such a case, the capital stock 
is ''pure velvet/' and costs nothing. Its future value, whatever 
it may be, is so much clear profit. The effect of such a financial 
policy is to burden the project with a capitalization which is 
far in excess of the actual cost of constructing the road. Com- 
paratively few projects will stand such over-capitalization. 
The apparent financial failure of many railroads, which have 
gone into the hands of receivers is due to their inability to 
make returns on an over-capitalization rather than because 
they could not earn enough to pay the legitimate cost of their 
construction. These features of financiering are really foreign 
to the engineer's work, but he should know that many projects 
which would return a handsome profit on an investment amount- 
ing only to the legitimate cost, will be rejected by capitalists 
because it is apparent that there is not enough ''velvet" 
in it. 

353. Item 2. Surveys and Engineering Expenses. The 
comparison of a large number of itemized reports on the cost 
of construction shows that the cost of the " engineering '' will 
average about 2% of the total cost of construction. This in- 
cludes the cost of surveys and the cost of laying out and super- 
intending the constructive work. The cost of mere surveying 
up to the time when construction actually commences has 
been variously quoted at $60, $75, and even $150 per mile. 
In exceptional cases the surveying for a few miles through some 
gorge might cost many times this amount, but $150 per mile 
may be considered an ordinary maximum for difl^cult country. 
On the other hand, much construction has been done over the 
western prairies after hasty surveys costing not much over 
$10 per mile. In the estimate given at the end of this chapter 
the cost of "engineering and office expenses'' is given at 5% of 
the cost of the construction work. The item then includes the 
cost of the very considerable amount of clerical work and 
superintendence incident to the expenditure of such a large 
sum of money. 

354. Item 3. Land and Land Damages. The cost of this 
item varies from the extreme, in which not only the land for 



§ 355. COST OF RAILROADS. 429 

right-of-way but also grants of public land adjoining the road 
are given to the corporation as a subsidy, to the other extreme, 
where the right-of-way can only be obtained at exorbitant 
prices. The width required is variable, depending on the 
width that may be needed for deep cuts or high fills, or the 
extra land required for yards, stations, etc. A strip of land 
1 mile long and 8.25 feet wide contains precisely 1 acre. An 
average widtl^ of 4 rods (66 feet), therefore, requires 8 acres per 
mile. On the Boston & Albany Railroad the expenditure 
assigned to "land and land damages" averages over $25000 
per mile. Of course this includes some especially expensive 
land for terminals and stations in large cities. Less than $300 
per mile was assigned to this item by an unimportant 18-mile 
road. 

355. Item 4. Clearing and Grubbing. The cost of this 
may vary from zero to 100% for miles at a time, but as an 
average figure it may be taken as about 3 acres per mile at a 
cost of say $50 per acre. The possibility of obtaining valuable 
timber, which may be utilized for trestles, ties, or otherwise, 
and the value of which may not only repay the cost of clearing 
and grubbing, but also some of the cost of the land, should not 
be forgotten. 

356. Item 5. Earthwork. This item also includes rock- 
work. The methods of estimating the cost of earthwork and 
rockwork have been discussed in Chapter III. The percentage 
of this item to the total cost is very variable. On a western 
prairie it might not be more than 5 to 10%. On a road through 
the mountains it will run up to 20 or 25%, and even more. 
The item also includes tunneling, which on some roads is a 
heavy item. 

357. Item 6. BRIDGES, TRESTLES, AND CULVERTS. This item 
will usually amount to 5 or 6% of the total cost of the road. 
In special cases, w^here extensive trestling is necessary, or 
several large bridges are required, the percentage will be much 
higher. On the Other hand, a road whose route avoids the 
watercourses may have very little except minor culverts. On 
the Boston & Albany the cost is given as $5860 per mile; on 
the Adirondack Railroad, $2845 per mile. Considering their 
relative character (double and single track), these figures are 
relatively what we might expect. 



130 



RAILROAD CONSTRUCTION. 



§ 358. 



358. Item 7. TRACKWORK. This item will be considered as 
including everything above subgrade, except as otherwise 
itemized. 

(a) Ballast. With an average width, for single track, of 
10 feet and an average depth of 15 inches, 2444 cubic yards of 
baJlast will be required. The Pennsylvania Railroad estimate is 
2500 yards of gravel per mile of single track. At an estimate 
of 60 c. per yard, this costs $1500 per mile. Broken-stone 
ballast must be filled out over the ends of the ties and there- 
fore more is required; 2800 cubic yards of broken stone at 
$1.25 per yard in place will cost $3500 per mile. 

(b) Ties. Ties cost anywhere from 80 c. down to 35 c. and 
even 25 c. At an aA^erage figure of 50 c, 2640 ties per mile 
will cost $1320 per mile of single track. The cheaper ties are 
usually smaller and more must be used per mile, and this tends 
to compensate the difference in cost. 

The following tabular form is convenient for reference: 



TABLE XV.— NUMBER OF CROSS TIES PER MILE. 



Number per 


Average spacing 


Number 


33' rail. 


center to center. 


per mile. 


22 


18.0 inches 


3520 


21 


18.9 " 


3360 


20 


19.8 *• 


3200 


19 


20.9 •• 


3040 


18 


22.0 •* 


2880 


17 


23.3 '• 


2720 


16 


24.75 •• 


2560 


15 


26.4 •• 


2400 


14 


28.3 " 


2240 


13 


30.5 " 


2080 



(c) Rails. The total weight of the rails used per mile may 
best be seen by the tabular form. 

A convenient and useful rule to remember is that the number 
of long tons (2240 lbs.) per mile of single track equals the weight 
of the rail per yard times V-. The rule is exact. For example, 
there are 3520 yards of rail in a mile of single track; at 70 lbs. 
per yard this equals 246400 lbs., or 110 long tons (exactly); 
but Vox V- = 110. 

Any calculation of the required weight of rail for a given 
weight of rolling-stock necessarily depends on the assumptions 
which are made regarding the support which the rails receive 
from the ties. This depends not only on the width and spacing 



§ 358. 



COST OF RAILROADS. 



431 



TABLE XVI. TONS PER MILE (wiTH COST) OF RAILS OF 

VARIOUS WEIGHTS. 





Tons 








Tons 






Weight 


(22401b.) 


Cost at 


Cost at 


Weight 


(22401b.) 


Cost at 


Cost at 


in lbs. 


per mile 


$26 per 


$30 per 


in lbs. 


per mile 


$26 per 


$30 per 


per yd. 


of single 
track., 


ton. 


ton. 


per yd. 


of single 
track. 


ton. 


ton. 


8 


12.571 


$326.86 


$377.14 


65 


102.143 


$2655.71 


$3064.29 


10 


15.714 


408.57 


471.43 


66 


103.714 


2696.57 


3111.43 


12 


18.857 


490.29 


565.71 


67 


105.286 


2737.43 


3158.59 


14 


22.000 


572.00 


660 . 00 


68 


106.857 


2778.29 


3205.79 


16 


25.143 


653.71 


754.20 


70 


110.000 


2860.00 


3300.00 


20 


31.429 


817.14 


942.86 


71 


111.571 


2900.86 


3347.14 


25 


39.286 


1021.43 


1178.57 


72 


113.143 


2941.71 


3394.29 


30 


47.143 


1225.71 


1414.29 


73 


114.714 


2982.57 


3441.43 


35 


55.000 


1430.00 


1650.00 


75 


117.857 


3064.29 


3535.71 


40 


62.857 


1634.29 


1885.71 


78 


122.571 


3186.86 


3677 . 14 


45 


70.714 


1838.57 


2121.43 


80 


125.714 


3268.57 


3771.43 


48 


75.429 


1961.14 


2262.86 


82 


128.857 


3350.29 


3865.71 


50 


78.571 


2042.86 


2357.14 


85 


133.571 


3472.86 


4007.14 


52 


81.714 


2124.57 


2451.43 


88 


138.286 


3595.43 


4148.57 


56 


88.000 


2288.00 


2640.00 


90 


141.429 


3677.14 


4242.86 


57 


89.571 


2328.86 


2687.14 


92 


144.571 


3758.86 


4337.14 


60 


94.286 


2451.43 


2828 . 57 


95 


149.286 


3881.43 


4478.57 


61 


95.857 


2492.29 


2875.71 


98 


154.000 


4004.00 


4620.00 


63 


99.000 


2574.00 


2970.00 


100 


157.143 


4085.71 


4714.29 



About two per cent. {2%) extra should be allowed for waste in cutting. 

of the ties (which are determinable), but also on the support 
which the ties receive from the ballast, which is not only very 
uncertain but variable. No general rule can therefore claim 
any degree of precision, but the following is given by the Bald- 
win Locomotive Works: " Each ten pounds weight per yard of 
ordinary steel rail, properly supported by cross- ties (not less 
than 14 per 30-foot rail), is capable of sustaining a safe load 
per wheel of 2240 pounds." For example, a consolidation loco- 
motive with 112600 lbs. on 8 driAcrs has a load of 14075 lbs. 
per wheel. This divided b}^ 2240 gives 6.28. According to the 
rule, the rails for such a locomotive should weigh at least 62.8 
lbs. per j^ard. 

(d) Splice-bars, track-bolts, and spikes. These are usually 
sold by the pound, except the patented forms of rail-joints, 
which are sold by the pair. In any case thej^ are subject to 
market fluctuations in price. As an approxim.ate value the 
following prices are quoted: Splice-bars, 1.35 c. per pound; 
track-bolts, 2.4 c; spikes, 1.75 c. The weight of the splice- 
bars will depend on the precise pattern adopted — its cross- 
section and length. 



432 



RAILROAD CONSTRUCTION. 



§358. 



In Table XVII are quoted from a catalogue of the Illinois 
Steel Co. the weights per foot of sections of angle-bars which 
they recommend for various weights of rail and which are de- 
signed to fit standard A. S. C. E. rail sections of those weights. 
The net weight of the angle-bars may be approximated by 
subtracting about 2.5% to 4% from the gross weight to allow 
for the bolt-holes. A deduction of 2.5% is usually about 
right for the heavier sections. Their recom.mendations regard- 
ing lengths of angle-bars do not include those for rails heavier 
than 50 pounds per yard. On the basis of a length of 23 inches 
for four-hole splices and of 33 inches for six-hole splices, the 
weights of splice-bars have been computed for the several 
styles of splices for heavier rails, allowing 2.5% for the holes. 
The lengths recommended for track bolts are those which will 
allow about i inch for the nutlock and for margin, except for 
the lighter rails. 



TABLE XVII. SPLICE-BARS FOR VARIOUS WEIGHTS OF RAILS. 



Weight 


Length 


Weight 


Weight 


Proper 


Proper size 


of 
rail. 


of 
angle-bar. 


per 
foot. 


of 
pair. 


size of 
track-bolt. 


of spikes. 


30 


21'' 


4.49 


15.1 


2i"Xf" 


4" XV 


35 


21" 


4.7 


15.9 


2r'X " 


4rxr 


40 


21" 


5.54 


18.8 


3 "X " 


5 "xr 


45 


21" 


6.3 


21.5 


3 "X^" 


5rxA" 


50 


21" 


6.97 


23.4 


3¥'Xt" 


srxfe" 


65 


23" 


7.5 


28.0 


31" X^" 


SV'XA" 


60 


23" 


8.4 


31.4 


3f"Xf" 


5V'X^." 


65 


f 23" 
\33" 


9.2 


34.4 


4 "Xf" 


5rx^" 


9.6 


51.5 


4rxr 


5rxi%" 


70 


/23" 


9.0 


33.6 


4 "XI" 


5rx^" 


\ 33" 


10.0 


53.6 


4 "Xf" 


5i"X^" 


15 


[23" 


10.68 


39.9 


4i"Xf" 


5i"XA" 


\33" 


11.9 


63.7 


4 "X-" 


5V'Xt%" 


80 


|23" 


10.61 


39.7 


4i"Xi" 


SV'Xt^" 


\33" 


14.65 


78.5 


^*r>^f7. 


5¥' X tS" 


85 


33" 


12.4 


66.4 


4>"X|" 


5V' X T^" or §'' 


90 


33" 


13.5 


72.3 


41" Xr 
41'' Xg" 


5i"XA" or r' 


95 


33" 


14.7 


78.7 


5rXA"orf" 


100 


33" 


15.78 


85.0 


41" Xs" 


5i"X^"or t" 



§ 358. 



COST OF RAILROADS. 



433 



TABLE XVIII. RAILROAD SPIKES. 



I 




Ties 24" between cen- 






Average 


ters, 4 spikes per tie, 


Suitable 


Size meas- 


number 


number per mile. 


weight of 


ured under 
head.. 


per keg of 
200 pounds 




rail. 










Pounds. 


Kegs. 




5V' X r' 


275 


7680 


38.40 


90 to 100 


5r'x^'' 


375 


5632 


28.16 


45 *• 100 


6" XA" 


400 


5280 


26.40 


40 • ' 56 


6" xr 


450 


4692 


23.46 


40 


4rxV' 
4- XV' 
4V' X^j" 


530 


3984 


19.92 


35 


600 


3520 


17.60 


30 


680 


3104 


15.52 


25 to 30 



TABLE XIX. TRACK-BOLTS. 

Average number in a keg of 200 pounds. 



Size of 


Square 


Hexagonal 


Suitable 


bolt. 


nut. 


nut. 


rail. 


3" Xg" 


366 


395 


40 pound 


3" X^" 


250 


270 




3i"X^" 


243 


261 




3rx^" 


236 


253 


50 


31" xi" 


229 


244 


55 to 60 


4" X^" 


222 


236 


65 " 70 


41" xi" 
3rXi" 


215 


228 


75 


170 


180 




31" xr 


165 


175 




4" XV' 
41" Xi" 


161 


170 




157 


165 


80 


4rxr 
4rxr 


153 


160 


85 


149 


156 


90 



434 RAILROAD CONSTRUCTION. § 358. 

(e) Track-laying. Much depends on the force of men em- 
ployed and the use of systematic methods; $528 per mile is 
the estimate employed by the Pennsylvania Railroad. $500 per 
mile is the estimate given in § 362. 

359. Item 8. Buildings and Miscellaneous structures. 
Except for rough and preliminary estimates, these items must 
be individually estimated according to the circumstances. The 
subitems include depots, engine-houses, repair-shops, water- 
stations, section- and tool-houses, besides a large variety of 
smaller buildings. The structures include tvirn-tables, cattle- 
guards, fencing, road-crossings, overhead bridges, etc. The 
detailed estimate, given in § 362, illustrates the cost of these 
smaller items. 

360. Item 9. INTEREST ON CONSTRUCTION. The amount 
of capital that must be spent on a railroad before it has begun 
to earn anything is so very large that the interest on the cost 
during the period of construction is a very considerable item. The 
amount that must be charged to this head depends on the cur- 
rent rate of money on the time required for construction and 
on the ability of the capitalists to retain their capital where 
it will be earning something until it is actually needed to pay 
the company^s obligations. Of course, it is not necessary to 
have the entire capital needed for construction on hand Avhen 
construction commences. Assuming money to be Avorth 6%, 
that the work of construction w^ill require one year, that the 
money mxay be retained where it wall earn something for an 
average period of six months after construction commences, 
or, in other words, it will be out of circulation six months before 
the road is opened for traffic and begins to earn its way, then 
we may charge 3% on the total cost of construction. 

361. Item 10. TELEGRAPH LINES. This evidently depends 
on the scale of the road and the magnitude of the business to 
be operated. In the following estimate it is given as $200 
per mile, which evidently is intended to apply to the business 
of a small road. 

362. Detailed estimate of the cost of a line of road. The fol- 
lowing estimate was given in the Engineering News of Dec. 27, 
1900, of the cost of the Duluth, St. Cloud, Glencoe ^ T^lankato 
Railroad, 157.2 miles long. 

The estimate is exactl}' as copied from the Engineering News. 
There are some numerical discrepancies. Item 26 should evi- 



§ 362c COST OF RAILROADS. 435 

dently be based on the sum of the first 25 items, and item 27 

on the sum of the first 26. The figures in parentheses ( ) are 
deduced from the figures given. 

1. Right-of-way; 1905.3 acres (12.12 acres per mile) @ $100 per 

acre , $190530 

2. Clearing and grubbing. 144 acres (0.916 acre per mile) @ $50 

per acre 7200 

3. Earth excavation. 1907590 cu. yds. (12135 cu. yds. per mile) 

@ 15 c 286138 

4. Rock excavation. 5100 cu. yds. (32.44 cu. yds. per mile) @ 80 c. 4080 
i Wooden-box culverts. 508300 ft. B.M. @ $30 per M. . $15249 

^' I Iron-pipe culverts. 879840 lbs. @ 3c. per lb 26395 41644 

{ Pile trestling . 4600 lin. ft. @ 35 c. per lin. ft 1610 

{ Timber trestling. 509300 ft. B.M. @ $30 per M 15279 16889 

j Bridge masonry: 5520 cu. yds. @, $8 per cu. yd 44160 

( Bridges, iron, 100 spans, 2000000 lbs. @ 4 c. per lb. . . 80000 124160 

8. Cattle-guards 8750 

9. Ties (2640 per mile). 419813 (159.02 miles) @ 35 c 146935 

10. Rails (70 lbs. per yd.): 110 tons per mile^ 17492.2 tons (159.02 

miles @$26 384797 

11. Rail sidings (70 lbs. per yd.) : 110 tons per mile, 3300 tons 

(30miles@ $26 85801 

12. Switch timbers and ties 3300 

13. Spikes: 5920 lbs. per mile. 1107040 (187 m.) @ 1.75. c. per lb. 19373 

14. Splice-bars. 2635776 lbs. @ 1.35 c. per lb 35583 

15. Track-bolts (2 to joint (?)): 188458.3 lbs. @ 2.4 c. per lb 4520 

16. Track-laying 187.2 miles @ $500 per mile 93600 

17. Ballasting- 2152 cu. yds. per mile, 402854 (187.2 m.) @ 60 c. . 241712 

18. Turn-out and switch furnishings 6450 

19. Road-crossings, 68040 ft. B.M. @ $30 per M 2041 

20. Section and tool-houses, 16 @ $800 12800 

21. Water-stations 15000 

22. Turn-tables, 6 @ $800 4800 

23. Depots, grounds, and repair-shops 78000 

24. Terminal grounds and special land damages 150000 

25. Fencing, 314 miles ($150 per mile) 47100 

26. Engineering and office expenses (5% of $1984458) 99222 

27. Interest on construction (3% of $2083680) 62510 

28. Rolling-stock ($5000 per mile) 786000 

29. Telegraph line: 157 miles @ $200 per mile 3140 

$3060340 
Average cost per mile ready for operation, $19467. 
Approximate cost of 130 miles from St. Cloud to Duluth, estimated at 

$23000 per mile. 
Approximate cost of entire line from Albert Lea to Duluth, 287.2 miles, 

$6050340 ($21060 per mile). 



PART II. 
RAILKOAD ECONOMICS. 



CHAPTER XVIII. 

INTRODUCTION. 

363. The magnitude of railroad business. The gross earnings 
of railroads for the year ending June 30, 1899, were over $1,300,- 
000,000. This is greater than the combined value of all the 
gold, silver, iron, wheat, and com produced by the country. 
The following figures (to the nearest million of dollars) gives 
the value of various crops for 1899, according to the current 
U. S. Yearbook of Agriculture: 



Gold 71 

Silver 33 

Iron 245 

Wheat 320 

Corn 629 



Oats 198 

Hay 412 

Coal 256 

Copper 104 

Lead 19 



About 929000 persons (about one eightieth of the population) 
were directly employed by the roads for a compensation of 
about $523,000,000. Probably 3,000,000 to 4,000,000 people 
were supported by this. Beside all these, probably 5,000,000 
employes were kept busy in occupations which are a more or 
less direct result of railroads, e.g.^ locomotive- and car-shops, 
rail-mills, etc. We may therefore estimate that perhaps 
20,000,000 people (or, say, one fourth of our population) are 
supported by railroads or by occupations which owe their 
chief existence to railroads. 

The ^^number of passengers carried 1 mile" was 14,591,327,613. 
Calling the population of the United States 75,000,000 for roimd 

436 



§ 364. ' INTRODUCTION. 437 

numbers, it means an average ride of 195 miles for every man, 
woman, and child. 

The 'Hons carried 1 mile'' were 123,667,257,153, or neariy 
1650 ton-miles per inhabitant. The payments made to the 
railroads averaged over $17 per inhabitant. 

Turning to a dark side of the picture, we find that the traffic 
was carried on at a cost of 7123 killed and 44620 injured. This 
averages one killed every hour and a quarter and one injured 
every twelve minutes. Of these large numbers, the '^passen- 
gers" comprised but 239 and 3442 respectively. The remainder 
were employes and '* others," the ^^ others" consisting largely 
of 'trespassers." 

The actual bona-fide cost of the railroads of the country 
cannot be accurately computed (as will be shown later), but 
the capital, as represented by stocks and bonds, represents 
$11,033,954,898, or about $147 per inhabitant. This is roughly 
about one sixth of the total national wealth. 

The above figures may give some idea of the magnitude of 
the interests involved in the operation of railroads. No single 
business in the country approaches it in capital involved, earn- 
ings, number of people affected, or effect on other business. 

364. Cost of transportation. The importance of railroads 
may be also indicated by their power of creating cheap trans- 
portation. Less than one hundred years ago local famine 
and overabundant harvests within a radius of a few miles 
were not unknown. When the transportation of goods depended 
on actual porterage by human beings, as has been the case 
but recently in the Klondike, the transportation of 100 lbs. 20 
miles might be considered an average day's labor. At $1 per 
day, this equals $1 per ton-mile. In 1899 the railroads trans- 
ported freight at an average cost to the public of 0.724 c. per 
ton per mile, and the feeding of Europe with wheat from Mani- 
toba has become a commercial possibility. In 1899 passengers 
paid an average charge of 1.925 c. per mile, and a trip of 1000 
miles inside of 24 hours is now common. 

365. Study of railroad economics — its nature and limitations. 
The multiplicity of the elements involved in most problems 
in railroad construction preclude the possibility of a solution 
which is demonstrably perfect. Barring out the compara- 
tively few cases in this country where it is difficult to obtain 
any practicable location, it may be said that a comparatively 



438 RAILROAD CONSTRUCTIONi § 365. 

low order of talent will suffice to locate anywhere a railroad 
over which it is physically possible to run trains. It may be 
very badly located for obtaining business, the ruling grades 
may be excessive, the alignment may be very bad, and the 
road may be a hopeless financial failure, and yet trains can be 
run. Among the infinite number of possible locations of the 
road, the engineer must determine the route which will give 
the best railroad property for the least expenditure of money — 
the road whose earning capacity is so great that after paying 
the operating expenses and interests on the bonds, the surplus 
available for dividends or improvements is a maximum. 

An unfortunate part of the problem is that even the blunders 
are not always readily apparent nor their magnitude. A de- 
fective dam or bridge will give way and every one realizes the 
failure, but a badly located railroad affects chiefly the finances 
of the enterprise by a series of leaks which are only perceptible 
and demonstrable by an expert, and even he can only say that 
certain changes would probably have a certain financial value. 

366. Outline of the engineer's duties. The engineer must 
realize at the outset the nature and value of the conflicting 
interests which are involved in variable amount in each possi- 
ble route. 

(a) The maximum of business must be obtained, and yet it may 
happen that some of the business may only be obtained by an 
extravagant expenditure in building the line or by building a 
line very expensive to operate. 

(b) The ruling grades should be kept low, and yet this may 
require a sacrifice in business obtained and also may cost more 
than it is worth. 

(c) The alignment should be made as favorable as possible; 
favorable alignment reduces the future operating expenses, 
but it may require a very large immediate outlay. 

(d) The total cost must be kept within the amount at which 
the earnings will make it a profitable investment. 

(e) The road must be completed and operated until the 
^^ normal" traffic is obtained and the road is self-supporting 
without exhausting the capital obtainable by the projectors ; 
for no matter how valuable the property may ultimately be- 
come, the projectors will lose nearly, if not quite, all they have 
invested if they lose control of the enterprise before it becomes 
a paying investment. 



§ 367. INTRODUCTION. 439 

Each new route suggested makes a new combination of the 
above conflicting elements. The engineer must select a route 
by first eliminating all lines which are manifestly impracticable 
and then gradually narrowing the choice to the best routes 
whose advantages are so nearly equal that a closer detailed 
comparison is necessary. 

The ruling grade and the details of alignment have a large 
influence on the operating expenses. A large part of this 
course of instruction therefore consists of a study of operating 
expenses under average normal conditions, and then a study 
of the effect on operating expenses of given changes in the align- 
ment. 

367. Justification of such methods of computation. It may 
be argued that the data on which these computations are based 
are so unreliable (because variable and to some extent non- 
computable) that no dependence can be based on the conclu- 
sions. This is true to the extent that it is useless to claim 
great precision in the computation of the value of any pro- 
posed change of alignment. Suppose, for example, it is com- 
puted that a given improvement in alignment will reduce the 
operating expenses of 20 trains per day by $1000 per year. 
Suppose the change in alignment may be made for $5000, which 
may be obtained at 5% interest. Even with large allowances 
for inaccuracy in the computation of the value, $1000, it evi- 
dently will be better to incur an additional interest charge of 
$250 than increase the annual operating expenses by $1000. 
Moreover, since traffic is almost sure to increase (and interest 
charges are generally decreasing), the advantage of the im- 
provement will only increase as time passes. On the other 
hand, if the improvement cannot be made except by an expen- 
diture of, say, $50000, the change would evidently be unjus- 
tifiable. When the interest on the first cost is practically 
equal to the annual operating value of the proposed improve- 
ment, there is evidently but little choice; no great harm can 
result from either decision, and the decision frequently will 
depend on the willingness to increase the total amount invested 
in the enterprise. 

To express the above question more generally, in every com- 
putation of the operating value of a proposed improvement, 
it may always be shown that the true value lies somewhere 
between some maximum and some minimum. Closer calcula- 



440 RAILROAD CONSTRUCTION. § 367. 

tions and more reliable data will narrow the range between 
these extreme values. According as the interest on the cost 
of the proposed improvement is greater or less than the mean 
of these limits, we may judge of its advisability. The range 
of the limits shows the uncertainty. If it lies outside of the 
limits there is no uncertainty, assuming that the limits have 
been properly determined. If well within the limits, either 
decision -will answer unless other considerations determine the 
question. And so, although it is not often possible to obtain 
precise values, we may generally reach a conclusion which is 
unquestionable. Even under the most unfavorable circum- 
stances, the computations, when made with the assistance of 
all the broad common sense and experience that can be brought 
to bear, will point to a decision which is much better than mere 
''judgment," which is responsible for ver}- many glaring and 
costly railroad blunders. In short. Railroad Economics means 
the application of systematic methods of work plus experience 
and judgment, rather than a dependence on judgment unsys- 
tematically formed. It makes no pretense to furnishing mechan- 
ical rules by which all railroad problems may be solved by any 
one, but it does give a general method of applying principles 
by which an engineer of experience and judgment can apply 
his knowledge to better advantage. To the engineer of limited 
experience the methods are invaluable; without such methods 
of work his opinions are practically worthless; with them 
his conclusions are frequently more sound than the unsystem- 
atically formed judgments of a man with a glittering record. 
But the engineer of great experience may use these methods 
to form the best opinions which are obtainable, for he can apply 
his experience to make any necessary local modifications in the 
method of solution. The dangers lie in the extremes, either 
recklessly applying a rule on the basis of insufficient data to 
an unwarrantable extent, or, disgusted with such evident 
unreliability, neglecting altogether such systematic methods of 
work. 



CHAPTER XIX. 

THE PROMOTION OF RAILROAD PROJECTS. 

368. Method of formation of railroad corporations. Many 
business enterprises, especially the smaller ones, are financed 
entirely by the use of money which is put into them directly 
in the form of stock or mere partnership interest. A railroad 
enterprise is frequently floated with a comparatively small 
financial expenditure on the part of the original promoters. 
The promoters become convinced that a railroad between A 
and B, passing through the intermediate towns of C and D, 
with others of less importance, will be a paying investment. 
They organize a company, have surveys made, obtain a charter, 
and then, being still better able (on account of the additional 
information obtained) to exploit the financial advantages of 
their scheme, they issue a prospectus and invite subscriptions 
to bonds. Sometimes a portion of these bonds are guaranteed, 
principal and interest, or perhaps the principal alone, by town- 
ships or by the national government. The cost of this pre- 
liminary work, although large in gross amount if the road is 
extensive, is yet but an insignificant proportion of the total 
amount involved. The proportionate amount that can be 
raised by means of bonds varies with the circumstances. In 
the early history of railroad building, when a road was pro- 
jected into a new country where the traffic possibilities were 
great and there was absolutely no competition, the financial 
success of the enterprise would seem so assured that no diffi- 
culty would be experienced in raising from the sale of bonds 
all the money necessary to construct and equip the road. But 
the promoters (or stockholders) must furnish all money for the 
preliminary expenses, and must make up all deficiencies be- 
tween the proceeds of the sale of the bonds and the capital needed 
for construction. 

''In theory, stocks represent the property of the responsible 
owners of the road, and bonds are an encumbrance on that 

441 



442 RAILROAD CONSTRUCTION. § 368. 

property. According to this theory, a railroad enterprise 
should begin with an issue of stock somewhere near the A^alue 
of the property to be created and no more bonds should be 
issued than are absolutely necessary to complete the enter- 
prise. Now it is not denied that there are instances in which 
this theory is followed out. In New England, for example, 
as well as in some of the Southern States, there are a few roads 
represented wholly by stock or very lightly mortgaged. But 
this theory does not conform to the general history of railway 
construction in the United States, nor is it supported by the 
figures that appear in the summary. The truth is, railroads 
are built on borrowed capital, and the amount of stock that is 
issued represents in the majority of cases the difference between 
the actual cost of the undertaking and the confidence of the 
public expressed by the amount of bonds it is willing to absorb 
in the ultimate success of the venture. '' * 

''The same general law obtains and has always obtained 
throughout the world, that such properties (as railways) are 
always built on borrowed money up to the limit of what is 
regarded as the positive and certain minimum value. The 
risk only — the dubious margin which is dependent upon sagac- 
ity, skill, and good management — is assumed and held by the 
company proper who control and manage the property.^' f 

369. The two classes of financial interests — the security and 
profits of each. From the above it may be seen that stocks, 
bonds, car-trust obligations, and even current liabilities repre- 
sent railroad capital. The issue of the bonds ''was one means 
of collecting the capital necessary to create the property against 
which the mortgage lies.'' The variation between these inter- 
ests lies chiefly in the security and profits of each. The current 
liabilities are either discharged 'or, as frequently happens, they 
accumulate until they are funded and thus become a definite 
part of the railroad capital. 

The growth of this tendency is shown in the following tabular 
form : 

The bonded interest has greater security than the stock, but 
less profit. The interest on the bonds must be paid before any 
money can be disbursed as dividends. If the bond interest 



* Henry C. Adams, Statistician, U. S. Int. Con. Commission, 
t A. M. Wellington, Economic Theory of Railway Location. 



§369. 



PROMOTION OF RAILROAD PROJECTS. 



443 



Capitalization of 


June 30, 1888. 


June 30, 1898. 


June 30, 1906. 


Railroads in the United 
States. 


Amount, 
millions. 


Per 
cent. 


Amount, 
millions. 


Per 
cent. 


Amount, 
millions. 


Per 
cent. 


Stocks 


3864 

3869 

396 


47.5 

47.6 

4.9 


5311 
5510 
1087 


44.6 

46.3 

9.1 


6804 
7767 
1101 


43 4 


Funded debt 

Current liabilities, etc. . 


49.6 
7.0 



is not paid, a receivership, and perhaps a foreclosure and sale 
of the road, is a probability, and in such case the stockholder's 
interests are frequently wiped out altogether. The bond- 
holder's real profit is frequently very different from his nomi- 
nal profit. He sometimes buys the bonds at a very considerable 
discount, which modifies the rate which the interest received 
bears to the amount really invested. Even the bondholder's 
security may suffer if his mortgage is a second (or fifth) mort- 
gage, and the foreclosure sale fails to net sufficient to satisfy 
all previous claims. 

On the other hand, the stockholder, who may have paid in 
but a small proportion of his subscription, may, if the venture 
is successful, receive a dividend which equals 50 or 100% of the 
money actually paid in, or, as before stated, his entire holdings 
may be entirely wiped out by a foreclosure sale. When the 
road is a great success and the dividends very large, additional 
issues of stock are generally made, which are distributed to the 
stockholders in proportion to their holdings, either gratuitously 
or at rates which give the stockholders a large advantage over 
outsiders. This is the process known as ^'watering." While 
it may sometimes be considered as a legitimate ^'salting down'' 
of profits, it is frequently a cover for dishonest manipulation of 
the money market. 

For the twelve years between 1887 and 1899 about two thirds 
of all the railroad stock in the United States paid no dividends, 
while of those that paid dividends the average rate varied 
from 4.96 to 5.74%. The year from June 30, 1898, to June 30, 
1899, was the most prosperous year of the group, and yet nearly 
60% of all railroad stock paid no dividend, and the average 
rate paid by those which paid at all was 4.96%. The total 
amount distributed in dividends was greater than ever before, 
but the average rate is the least of the above group because many 
roads, which had passed their dividends for many previous 



I 



444 RAILROAD CONSTRUCTION. § 369c 

years, distinguished themselves by declaring a dividend, even 
though small. During that same period but 13.35% of the 
stock paid over 6% interest. The total dividends paid amounted 
to but 2.01% of all the capital stock, while investments ordi- 
narily are expected to yield from 4 to 6% (or more) according 
to the risk. Of course the effect of ''watering" stock is to 
decrease the nominal rate of dividends, but there is no dodging 
the fact that, watered or not, even in that year of ''good times," 
about 60% of all the stock paid no dividends. Unfortunately 
there are no accurate statistics showing how much of the stock 
of railroads represents actual paid-in capital and how much 
is "water." The great complication of railroad finances and 
the dishonest manipulation to which the finances of some rail- 
roads have been subjected would render such a computation 
practically worthless and hopelessly unreliable now. 

During the year ending June 30, 1898 (which may in general 
be considered as a sample), 15.82% of the funded debt paid no 
interest. About one third of the funded debt paid between 
4 and 5% interest, which is about the average which is paid. 

The income from railroads (both interest on bonds and divi- 
dends on stock) may be shown graphically by diagrams, such 
as are given in the annual reports of the Interstate Commerce 
Commission. They show that while railroad investments are 
occasionally very profitable, the average return is less than 
that of ordinary investments to the investors. The indirect 
value of railroads in building up a section of country is almost 
incalculable and is worth many times the cost of the roads. 
It is a discouraging fact that very few railroads (old enough to 
have a history) have escaped the experience of a receivership, 
with the usual financial loss to the then stockholders. But 
there is probably not a railroad in existence which, however 
much a financial failure in itself, has not profited the community 
more than its cost. 

370. The small margin between profit and loss to projectors. 
When a railroad is built entirely from the funds furnished by 
its promoters (or from the sale of stock) it will generally be a 
paying investment, although the rate of payment may be very 
small. The percentage of receipts that is demanded for actual 
operating expenses is usually about 67%. The remainder will 
usually pay a reasonable interest on the total capital involved. 
But the operating expenses are frequently 90 and even 100% of 



§ 371. PROMOTION OF RAILROAD PROJECTS. 445 

the gross receipts. In such cases even the bondholders do not 
get their due and the stockholders have absolutely nothing. 
Therefore the stockholder's interest is very speculative. A 
comparatively small change in the business done (as is illus- 
trated numerically in § 372) will not only wipe out altogether the 
dividend — taken from the last small percentage of the total 
receipts and which may equal 50% or more of the capital stock 
actually paid in — but it may even endanger the bondholders' 
security and cause them to foreclose their mortgage. In such 
a case the stockholders' interest is usually entirely lost. It 
does not alter the essential character of the above-stated rela- 
tions that the stockholders sometimes protect themselves 
somewhat by buying bonds. By so doing they simply decrease 
their risk and also decrease the possible profit that might result 
from the investment of a given total amount of capital. 

371. Extent to which a railroad is a monopoly. It is a popu- 
lar fallacy that a railroad^ when not subject to the direct com- 
petition of another road, has an absolute monopoly — that it 
controls ^^all the traffic there is" and that its income will be 
practically independent of the facilities afforded to the public. 
The growth of railroad traffic, like the use of the so-called 
necessities or luxuries of life, depends entirely on the supply 
and the cost (in money or effort) to obtain it. A large part of 
railroad traffic belongs to the unnecessary class — such as travel- 
ing for pleasure. Such traffic is very largely affected by mere 
matters of convenience, such as well-built stations, convenient 
terminals, smooth track, etc. The freight traffic is very largely 
dependent on the possibility of delivering manufactured articles 
or produce at the markets so that the total cost of production 
and transportation shall not exceed the total cost in that 
same market of similar articles obtained elsewhere. The crea- 
tion of facilities so that a factory or mine may successfully 
compete with other factories or mines will develop such traffic. 
The receipts from such a traffic may render it possible to still 
further develop facilities which will in return encourage further 
business. On the other hand, even the partial withdrawal of 
such facilities may render it impossible for the factory or mine 
to compete successfully with rivals; the traffic furnished by 
them is completely cut off and the railroad (and indirectly the 
whole community) suffers correspondingly. The ''strictly 
necessary" traffic is thus so small that few railroads could pay 



446 



RAILROAD CONSTRUCTION. 



§372. 



their operating expenses from it. The dividends of a road 
come from the last comparatively small percentage of its revenue, 
and such revenue comes from the '^ unnecessary'' traffic which 
must be coaxed and which is so easily affected by apparently 
insignificant '^ conveniences." 

372. Profit resulting from an increase in business done; loss 
resulting from a decrease. In a subsequent chapter it will 
be shown that a large portion of the operating expenses are 
independent of small fluctuations in the business done and that 
the operating expenses are roughly two thirds of the gross 
revenue. Assume that by changes in the alignment the business 
obtained has been increased (or diminished) 10%. Assume for 
simplicity that the operating expenses on the revised track 
are the same as on the route originally planned; also that the 
cost of the track is the same and hence the fixed charges are 
assumed to be constant for all the cases considered. Assume 
the fixed charges to be 28%. The additional business, when 
carried in cars otherwise but partly filled will hardly increase 
the operating expenses by a measurable amount. When 
extra ears or extra trains are required, the cost will increase 
up to about 60% of the average cost per train mile. We may 
say that 10% increase may in general be carried at a rate of 
40% of the average cost of the traffic. A reduction of 10% 
in traffic may be assumed to reduce expenses a similar amount. 
The effect of the change in business will therefore be as follows : 





Business increased 10%. 


Business decreased 10%. 


Operating exp. = 67 
Fixed charges = 28 


67(1 + 10%X40%)= 69.68 
28.00 


67(1 - 10% X 40%)= 64.32 
28 . 00 








95 
Total income. . . 100 


97.68 
Income 110.00 


92.32 
Income 90 . 00 


Available for divi- 
dends 5 


Available for divi- 
dends 12.32 


Deficit 2.32 



In the one case the increase in business, which may often 
be obtained by judicious changes in the alignment or even by 
better management without changing the alignment, more than 
doubles the amount available for dividends. In the other case 
the profits are gone, and there is an absolute deficit. The 
above is a numerical illustration of the argument, previously 



§ 373, PROMOTION OF RAILROAD PROJECTS. 447 

stated, of the small margin between profit and loss to the original 
projectors. 

373. Estimation of probable volume of traffic and of probable 
growth. Since traffic and traffic facilities are mutually inter- 
dependent and since a large part of the normal traffic is merely 
potential until the road is built, it follows that the traffic of a 
road will not attain its normal volume until a considerable 
time after it is opened for operation. But the estimation even 
of this normal volume is a very uncertain problem. The esti- 
mate may be approached in three ways: 

1st. The actual gross revenue derived by all the railroads 
in that section of the country (as determined by State or U. S. 
Gov. reports) may be divided by the total population of the 
section and thus the average annual expenditure per head of 
population may be determined. A determination of this value 
for each one of a series of years will give an idea of the normal 
rate of growth of the traffic. Multiplying this annual contri- 
bution by the population which may be considered as tributary 
gives a valuation of the possible traffic. Such an estimate is 
unreliable (a) because the average annual contribution may not 
fit that particular locality, (6) because it is very difficult to 
correctly estimate the number of the true tributary population 
especially when other railroads encroach more or less into the 
territory. Since a rough value of this sort may be readily 
determined, it has its value as a check, if for nothing else. 

2d. The actual revenue obtained by some road whose 
circumstances are as nearly as possible identical with the road 
to be considered may be computed. The weak point consists 
in the assumption that the character of the two roads is identical 
or in incorrectly estimating the allowance to be made for ob- 
served differences. The method of course has its value as a 
check. 

3d. A laborious calculation may be made from an actual 
study of the route — determining the possible output of all 
factories, mines, etc., the amount of farm produce and of lumber 
that might be shipped, with an estimate of probable passenger 
traffic based on that of like towns similarly situated. This 
method is the best when it is properly done, but there is always 
the danger of leaving out sources of income — both existent 
and that to be developed by traffic facilities, or, on the other 
hand^ of overestimating the value of expected traffic. In the 



448 



RAILROAD CONSTRUCTION. 



§373. 



following tabular form are shown the population, gross re- 
ceipts, receipts per head of population, mileage, earnings per 
mile of line operated, and mileage per 10,000 of population for 
the whole United States. It should be noted that the values 
are only averages, that individual variations are large, and that 
only a very rough dependence may be placed on them as applied 
to any particular case. 









Receipts 




Earnings 


Mileage 




Population 


Gross 


per head 




per mile 


per 


Year. 


(estimated). 


receipts. 


of popu- 
lation. 


Mileage, t 


of line 
operated. 


10,000 
popula- 
tion.t 


1888.. . 


60,100,000 


$910,621,220 


$15.15 


136,884 


$6653 


24.94 


1889.. . 


61,450,000 


964,816,129 


15.81 


153,385 


6290 


25.67 


1890.. . 


*62,801,571 


1051,877,632 


16.75 


156,404 


6725 


26.05 


1891... 


64.150.000 


1096,701.395 


17.10 


161,275 


6801 


26.28 


1892.. . 


65.500.000 1 1171,407,343 


17.89 


162.397 


7213 


26.19 


1893.. . 


68,850,000 


1220,751,874 


18.26 


169,780 


7190 


26.40 


1894.. . 


68.200,000 


1073.361,797 


15.74 


175,691 


6109 


26.20 


1895.. . 


69.550.000 


1075,371,462 


15.46 


177,746 


6050 


25.97 


1896.. . 


70,900,000 


1150,169,376 


16.22 


181,983 


6320 


25.78 


1897.. . 


72.350,000 


1122,089,773 


15.53 


183,284 


6122 


25.53 


1898.. . 


73,600,000 


1247,325,621 


16.95 


184,648 


6755 


25.32 


1899... 


74.950,000 


1313,610,118 


17.53 


187,535 


7005 


25.25 


1900 . . 


*76,295,220 1487,044,814 


19.49 


192,556 


7722 


25.44 


1901 . . 


77,600,000 1588,526,037 


20.47 


195,562 


8123 


25.52 


1902 . . 


78,900,000 1726,380,267 


21.88 


200,155 


8625 


25.76 


1903 . . 


80,200,000 1900,846,907 


23.70 


205,314 


9258 


26.03 


1904 . . 


81,500.000 1975,174 091 


24.23 


212,243 


9306 


26.34 


1905 . . 


82,800.000 i 2082 482,406 


25.15 


216,974 


9598 


26.44 


1906 . . 


84,100,000 i 2325.765,167 


27.65 


222,340 


10460 


26.78 



* Actual. t Excludes a small percentage not reporting "gross receipts." 
X Actual mileage. 

The probable growth in traffic, after the traffic has once 
attained its normal volume, is a small but almost certainquantity. 
In the above tabular form this is indicated by the gradual 
growth in ''receipts per head of population" from 1888 to 
1893. Then the sudden drop due to the panic of 1893 is clearly 
indicated, and also the gradual growth in the last few years. 
Even in England, where the population has been nearly station- 
ary for many years, the growth though small is unmistakable. 
On the other hand the growth in some of the Western States 
has been very large. For example, the gross earnings per head 
of population in the State of Iowa increased from $1.42 in 1862 
to $10.00 in 1870, and to $19.46 in 1884. 

There will seldom be any justification in building to accommo- 
date a larger business than what is ''in sight." Even if it 
could be anticipated with certainty that a large increase in 



§ 374. PROMOTION OF RAILROAD PROJECTS. 449 

business would come in ten years, there are many reasons why 
it would be unwise to build on a scale larger than that required 
for the business to be immediately handled. Even though it 
may cost more in the future to provide the added accommo- 
dations (e.g. larger terminals, engine-houses, etc.), the extra 
expense will be nearly if not quite offset by the interest saved 
by avoiding the larger outlay for a period of years which may 
often prove much longer than was expected. A still more im- 
portant reason is the avoidance of uselessly sinking money at 
a time when every cent may be needed to insure the success 
of the enterprise as a whole. 

374. Probable number of trains per day. Increase with 
growth of traffic. The number of passenger trains per day 
cannot be determined by dividing the total number of passengers 
estimated to be carried per day by the capacity of the cars 
that can be hauled by one engine. There are many small 
railroads, running three or four passenger trains per day each 
way, which do not carry as many passengers all told as are 
carried on one heavy train of a trunk line. But because the 
bulk of the passenger traffic, especially on such light-traffic 
roads, is '' unnecessary" traffic (see § 371) and must be encouraged 
and coaxed, the trains must be run much more frequently 
than mere capacity requires. The minimum number of passen- 
ger trains per day on even the lightest-traffic road should be 
two. These need not necessarily be passenger trains exclusively. 
They may be mixed trains. 

The number required for freight service may be kept more 
nearly according to the actual tonnage to be moved. At least 
one local freight will be required, and this is apt to be considerably 
within the capacity of the engine. Some very light -traffic 
roads have little else . than local freight to handle, and on such 
there is less chance of economical management. Roads with 
heavy traffic can load up each engine quite accurately according 
to its hauling capacity and the resulting economy is great. Fluc- 
tuations in traffic are readily allowed for b}^ adding on or drop- 
ping off one or more trains. Passenger trains must be run on 
regular schedule, full or empty. Freight trains are run by 
train-despatcher's orders. A few freight trains per day may be 
run on a nominal schedule, but all others will be run as extras. 
The criterion for an increase in the number of passenger trains 
is impossible to define by set rules. Since it should always 



450 RAILROAD CONSTRUCTION. § 375. 

come before it is absolutely demanded by the train capacity 
being overtaxed, it may be said in general terms that a train 
should be added when it is believed that the consequent in- 
crease in facilities will cause an increase in traffic the value of 
which will equal or exceed the added expense of the extra train. 

375. Effect on traffic of an increase in facilities. The term 
facilities here includes everything which facilitates the transport 
of articles from the door of the producer to the door of the 
consumer. As pointed out before, in many cases of freight 
transport, the reduction of facilities below a certain point Avill 
mean the entire loss of such traffic owing to local inability to 
successfully compete with more favored localities. Sometimes 
owing to a lack of facilities a railroad company feels compelled 
to pay the cartage or to make a corresponding reduction on 
what would normally be the freight rate. In competitive freight 
business such a method of procedure is a virtual necessity in 
order to retain even a respectable share of the business. Even 
though the railroad has no direct competitor, it must if possible 
enable its customers to meet their competitors on even terms. 
In passenger business the effect of facilities is perhaps even 
more marked. The pleasure travel wnll be largely cut down 
if not destroyed. It is on record that a railroad company 
once ordered the manager of a station restaurant to largely 
increase the attractions at that restaurant (as a method of 
attracting traffic) and agreed to pay the expected resulting 
loss. The net result was not only a large increase in railroad 
business (as was expected), but even an increase in the profits 
of the restaurant. 

376. Loss caused by inconvenient terminals and by stations 
far removed from business centers. This is but a special case 
of the subject discussed just in the preceding paragraph. The 
competition once existing between the West Shore and the 
New York Central was hopeless for the West Shore from the 
start. The possession of a terminal at the Grand Central 
Station gave the New York Central an advantage over the West 
Shore vrith its inconvenient terminal at Weehawken which 
could not be compensated by any obtainable advantage by 
the West Shore. This is especially true of the passenger busi- 
ness. The through freight business passing through or termi- 
nating at New York is handled so generally by means of floats 
that the disadvantage in this respect is not so great. The 



§ 376. PROMOTION OF RAILROAD PROJECTS. 451 

enormous expenditure (roughly $10,000,000) made by the 
Pennsylvania R. R., on the Broad Street Station (and its ap- 
proaches) in Philadelphia, a large part of which was made in 
crossing the Schuylkill River and running to Cit}^ Hall Square, 
rather than retain their terminal in West Philadelphia, is an 
illustration of the policy of a great road on such a question. 
The fact that the original plan and expenditure has been very 
largely increased since the first construction proves that the 
management has not only approved the original large outlay, 
but saw the wisdom of making a very large increase in the ex- 
penditure. 

The construction of great terminals is comparatively infrequent 
and seldom concerns the majority of engineers. But an engineer 
has frequently to consider the question of the location of a 
way station with reference to the business center of the town. 
The following points may (or may not) have to be considered, 
and the real question consists in striking a proper balance 
between conflicting considerations. 

(1) During the early history of a railroad enterprise it is 
especially needful to avoid or at least postpone all expenditures 
which are not demonstrably justifiable. 

(2) The ideal place for a railroad station is a location im- 
mediately contiguous to the business center of the town. The 
location of the station even one fourth of a mile from this may 
result in a loss of business. Increase this distance to one mile 
and the loss is very serious. Increase it to five miles and the 
loss approaches 100%. 

(3) The cost of the ideal location and the necessary right 
of way may be a very large sum of money for the new enterprise. 
On the other hand the increase in property values and in the 
general prosperity of the town, caused by the railroad itself, 
will so enhance the value of a more convenient location that its 
cost at some future time will generally be extravagant if not 
absolutely prohibitory. The original location is therefore under 
ordinary conditions a finality. 

(4) To some extent the railroad will cause a movement of 
the business center toward it, especially in the establishment 
of new business, factories, etc., but the disadvantages caused 
to business already established is permanent. 

(5) In any attempt to compute the loss resulting from a 
location at a given distance from the business center it must be 



452 RAILROAD CONSTRUCTION. § 376. 

recognized that each problem is distinct in itself and that any 
change or growth in the business of the town changes the amount 
of this loss. 

The argument for locating the station at some distance from 
the center of the town may be based on (a) the cost of right 
of way, thus involving the question of a large initial outlay, 
(6) the cost of very expensive construction (e.gr. • bridges), 
again involving a large initial outlay, (c) the avoidance of ex- 
cessive grade into and out of the town. It sometimes happens 
that a railroad is following a line which would naturally cause 
it to pass at a considerable elevation above (rarely below) 
the town. In this case there is to be considered not only the 
possible greater initial cost, but the even more important increase 
in operating cost due to the introduction of a very heavy grade. 
To study such a case, compute the annual increase in operating 
expenses due to the additional grade, curvature, and distance; 
add to this the annual interest on the increased initial cost 
(if any) and compare this sum with the estimated annual loss 
due to the inconvenient location. The estimation of the increase 
in operating expenses is discussed in a subsequent chapter. 
The loss of business due to inconvenient location can only be 
guessed at. Wellington says that at a distance of one mile 
the loss would average 25%, with upper and lower limits of 
10 and 40%, depending on the keenness of the competition 
and other modifying circumstances. For each additional mile 
reduce 25% of the preceding value. While such estimates are 
grossly approximate, yet with the aid of sound judgment they 
are better than nothing and may be used to check gross errors. 

377. General principles which should govern the expenditure 
of money for railroad purposes. It T\dll be sho^vn later that 
the elimination of grade, curvature, and distance have a positive 
money value ; that the reduction of ruling grade is of far greater 
value ; that the creation of facilities for the handling of a large 
traffic is of the highest importance and yet the added cost of 
these improvements is sometimes a large percentage of the 
cost of some road over which it would be physically possible 
to run trains between the termini. 

The subsequent chapters will be largely devoted to a discussion 
of the value of these details, but the general principles governing 
the expenditure of money for such purposes may be stated as 
follows: 



§ 377. PROMOTION OF RAILROAD PROJECTS. 453 

1. No money should be spent (beyond the unavoidable 
minimum) unless it may be shown that the addition is in itself 
a profitable investment. The additional sum may not wreck 
the enterprise and it may add something to the value of the 
road, but unless it adds more than the improvement costs it is 
not justifiable. 

2. If it may be positively demonstrated that an improvement 
will be more valuable to the road than its cost, it should certainly 
be made even if the required capital is obtained with difficulty. 
This is all the more necessary if the neglect to do so will per- 
manently hamper the road with an operating disadvantage 
which will only grow worse as the traffic increases. 

3. This last principle has two exceptions: (a) the cost of 
the improvement may wreck the whole enterprise and cause 
a total loss to the original investors. For, unless the original 
promoters can build the road and operate it until its stock 
has a market value and the road is beyond immediate danger 
of a receivership, they are apt to lose the most if not all of 
their investment; (b) an improvement which is very costly 
although unquestionably wise may often be postponed by means 
of a cheap temporary construction. Cases in point are found 
at many of the changes of alignment of the Pennsylvania R. R., 
the N. Y., N. H. & H. R. R., and many others. While some of 
the cases indicate faulty original construction, at many of the 
places the original construction was wise, considering the then 
scanty traffic, and now the improvement is wise considering 
the great traffic. 



CHAPTER XX. 

OPERATING EXPENSES. 

378. Distribution of gross revenue. When a railroad com- 
prises but one single property, owned and operated by itself, 
the distribution of the gross revenue is a comparatively simple 
matter. The operating expenses then absorb about two thirds 
of the gross revenue; the fixed charges (chiefly the interest on 
the bonds) require about 25 or 30% more, lea^dng perhaps 3 
to 8% (more or less) available for dividends. A recent report 
on the Fitchburg R. R. shows the following: 

Operating expenses $5,083,571 69. 1% 

Fixed charges 1,567,640 21.3% 

Available for dividends, surplus, or per- 
manent improvements 708,259 9.6% 

Total revenue $7,359,470 100.0% 

But the financial statements of a large majority of the railroad 
corporations are by no means so simple. The great consolida- 
tions and reorganizations of recent years have been effected 
by an exceedingly complicated system of leases and sub-leases, 
purchases, ^^ mergers," etc., whose forms are various. Railroads 
in their corporate capacity frequently ovm stocks and bonds 
of other corporations (railroad properties and otherwise) and 
receive, as part of their income, the dividends (or bond interest) 
from the investments. 

In consequence of this complication, the U. S. Interstate 
Commerce Commission presents a ^'condensed income account^' 
oi which the following is a sample (1899): 

Gross earnings from operation (received by 

station-agents, etc) $1,313,610,118 

Less operating expenses (fuel, wages, etc.) 856,968,999 

Income from operation 456,641,119 

Income from other sources (lease of road, stocks, 

bonds, etc.) 148.713,983 

Total income 605,355.102 

454 



§ 378. OPERATING EXPENSES. 455 

Total deductions from income (interest, rents for 

lease of road, taxes, etc.) 441 200,289 

Net income 164,154,813 

Total dividends (including *'other payments"). • 111,089,936 

Surplus from operations 53,064,877 

In the above account an item of income {e.g. lease of road) 
reported by one road will be reported as a ^'deduction from 
income'' by the road which leases the other. 

The above statement may be reduced to an income account 
of all the railways considered as one system. We then have 

Operating expenses $856,968,999 

Salaries and maintenance of leased lines 
(really operating expenses, but con- 
sidered above as fixed charges against 
the leasing lines) 595,192 

857,564,191 64.1% 

Net interest and taxes 295,098,014 22 . 0% 

Available for dividends, adjustments, 

and improvements 186,992,909 13.9% 

1,339,655,114 100.0% 

Gross earnings from operation 1,313,610,118 

Clear income from investments (i.e., 
the balance of intercorporate pay- 
ments and receipts on corporate in- 
vestments) 26,044,996 

1.339,655,114 

Of the $186,992,909, the amount disbursed as dividends to 
outside stockholders (besides that paid to railroads in their 
corporate capacity) was $94,273,796. This left a balance of 
$92,719,113 ''available for adjustments and improvements." 
Of this, part was spent in permanent improvements, part was 
advanced to cover deficits in the operation of weak lines and 
more than half was left as ''surplus/' i.e. working capital. 

The percentages of the gross revenue which are devoted to 
operating expenses, fixed charges, and dividends are not neces- 
sarily an indication of creditable management or the reverse. 
Causes utterly beyond the control of the management, such 
as the local price of coal, may abnormally increase certain 
items of expense, while ruinous competition may cut down the 
gross revenue so that little or nothing is left for dividends. 
A favorable location will sometimes make a road prosperous 



456 RAILROAD CONSTRUCTION. § 379. 

in spite of bad management. On the other hand, the highest 
grade of skill will fail to keep some roads out of the hands of 
a receiver. 

379. Fourfold distribution of operating expenses. The 
distribution of operating expenses here used is copied from the 
method of the Interstate Commerce Commission. The aim is to 
divide the expenses into groups which are as mutually indepen- 
dent and distinct as possible — although, as will be seen later, 
a change in one item of expense will variously affect other 
items. The groups are: 

Average value. 

1. Maintenance of way and structures 20.996% 

The values for ten years have an extreme range of 
about 2.7%. The subdivisions of this group and of 
the others will be given later. 

2. Maintenance of equipment 18 . 925% 

Growth in ten years over 5%. The tendency has been 
for this item to grow larger, not only in absolute amount 
but in percentage of total expenditure. 

3. Conducting transportation 65.946% 

This item has been growing relatively less. During 
(and immediately after) the panic of 1893. the mainte- 
nance of way and of equipment was made as small as 
possible, which made the cost of conducting transpor- 
tation relatively larger. During the recent more pros- 
perous years deficiencies of equipment have been made 
up, making this item relatively less. 

4. General expenses 4 . 133% 

A nearly constant item. 

100.000% 

The above percentages represent the averages given by the 
reports for the ten years from 1897 to 1906 inclusive. 

380. Operating expenses per train-mile. The reports of the 
U. S. Interstate Commerce Commission give the average cost 
per train-mile for every railroad in the United States. Although 
there are wide variations in these values, it is remarkable that 
the very large majority of roads give values which agree to 
within a small range, and that within this range are found not 
only the great trunk lines with their enormous train mileage, 
but also roads Vith very light traffic. 

In the following tabular form is shown a statement taken from 
the report for 1898 of ten of the longest railroads in the United 
States and, in comparison ^^ ith them, a corresponding statement 



380. 



OPERATING EXPENSES. 



457 



regarding ten more roads selected at random, except in the 
respect that each had a mileage of less than 100 miles. Al- 
though the extreme variations are greater, yet there is no very 
marked difference in the general values for operating expenses 
per train-mile, or in the ratio of expenses to earnings. The 
averages for the ten long roads agree fairly well with the averages 
for the whole country, but there would be no trouble (as is 
shown by some of the individual cases) in finding another group 
of ten short roads giving either greater or less average values than 
those given. And yet the tendency to uniform values, regard- 
less of the mileage, is very striking. 



No. in 
report. 



Whole United States. 



Mileage. 



186,396 



Operat- 
ing ex- 
penses 
per train 
mile. 



0.956 



Ratio- 



65.58 



71 
1465 
1443 
1879 
1142 
1436 
1405 
1560 
1495 
1264 



Canadian Pacific 

C.,M. &St.P 

C.,B.,&Q 

Southern Pacific 

Southern ! 

Chicago & Northwestern . 

A.,T. &S. F 

Northern Pacific 

Great Northern 

Illinois Central 



Average of ten . 



6,568 
6,191 
5,860 
5,426 
5,232 
5,086 
4,565 
4,524 
3,860 
3,807 



0.854 
0.883 
0.881 
1.320 
0.809 
0.885 
0.917 
1.177 
1.101 
.764 



0.969 



58.31 
58.94 
60.87 
58.70 
65.32 
63.35 
67.59 
46.81 
46.97 
63.56 



59.04 



7 

105 

167 

234 

888 

1074 

1284 

1540 

1812 

1979 



Bennington & Rutland. 

Mont. & Wells R 

Balto. & Del. Bay 

Cent. N. Y. &W 

Man. & N. E 

Farmv. & Powh 

Lex. & East 

Manistique 

Wh. & Bl. River Val. . . 
No. Pac. Coast 



Average of ten . 



59 
44 
45 
63 
99 
93 
94 
60 
64 
88 



0.582 
0.828 
1.098 
0.454 
0.739 
0.781 
0.975 
1.162 
.799 
.769 



0.819 



71.42 
83.96 
102.83 
91.17 
54.49 
76.22 
68.46 
69.01 
53.08 
66.58 



73.72 



Thefiuctuationsof the average cost per train mile for several 
years past may be noted from the following tabular form. 

The enforced economies after the panic of 1893 are well 
shown. The reduction generally took the form of a lowering 
of the standards of maintenance of way and of maintenance of 



J 



458 



OPERATING EXPENSES. 



§381. 



Year. 


Average cost per 
train-mile 
in cents. 


Year. 


Average cost per 
train-mile 
in cents. 


1890 


96.006 
95.707 
96 . 580 
97.272 
93.478 
91.829 
93 . 838 
92.918 
95.635 


1899 


98 390 


1891 


1900 


107 288 


1892 


1901 


112 292 


1893 


1902 


117 960 


1894 


1903 


126 604 


1895 

1896 


1904 

1905 


131.375 
132 140 


1897 

1898 


1906 


137.060 



equipment. The marked advance since 1895 is partly due to 
the necessity for restoring the roads to proper condition, re- 
plenishing worn-out equipment and providing additional equip- 
ment to handle the greatly increased volume of business. The 
recent advance is chiefly due to the increase in wages and the 
generally increased cost of supplies. 

In looking over the list, it may be noted that the cases where 
the operating expenses per train-mile and the ratio of expenses 
to earnings vary very greatly from the average are almost 
invariably those of the very small roads or of '^junction roads" 
where the operating conditions are abnormal. For example, one 
little road, with a total length of 13 miles and total annual opera- 
ting expenses of $5842, spent but 22 Jc. per train-mile, which pre- 
cisely exhausted its earnings. This precise equality of earnings 
and expenses suggests jugglery in the bookkeeping. As another 
abnormal case, a road 44 miles long spent S3. 81 per train-mile, 
which was nearly fourteen times its earnings. In another case a 
road 13 miles long earned $7.76 per train-mile and spent $6.03 
(78%) on operating expenses, but the fixed charges were abnor- 
mal and the earnings were less than half the sum of the operating 
expenses and fixed charges. The normal case, even for the 
small road, is that the cost^per train-mile and the ratio of operat- 
ing expenses to earnings will agree fairly well with the average, 
and when there is a marked difference it is generally due to 
some abnormal conditions of expenses or of earning capacity. 

381. Reasons for uniformity in expenses per train- mile. 
The chief reason is that, although on the heavy-traffic road 
everything is kept up on a finer scale, better roadbed, heavier 
rails, better rolling stock, more employees, better buildings, 
stations, and terminals, etc., yet the number of trains is so much 
greater that the divisor is just enough larger to make the average 



§ 381. OPERATING EXPENSES. 459 

cost about constant. This is but a general statement of a fact 
which will be discussed in detail under the different items of 
expense. 

382. Detailed classification of expenses with ratios to the 
total expense. The Interstate Commerce Commission now 
publishes each year a classification with detailed summation 
for the cost of each item. These summations are made up 
from reports furnished by railroads which have (in the reports 
recently made) represented over 99% of the total trafHc han- 
dled. In the annexed tabular form (Table XX) are shown the 
percentages which each item bears to the total. The character 
of the changes from year to year in these ratios is very instruct- 
ive and will be considered in the detailed discussion of the 
items which will follow. 

Table XX is compiled from the Interstate Commerce Com- 
mission reports for the several years mentioned. 

383. Elements of the cost (with variations and tendencies) 
of the various items. The I. C. C. report for the year ending 
June 30, 1895, was the first to include the distribution of 
expenses according to the present classification. The items as 
given are reliable and may be utilized, as far as any such com- 
putations are to be depended on, in estimating future expense?. 
A great deal of very interesting and instructive information may 
be derived from a study of the variations of these items, but the 
chief purpose of this discussion is to point out those elements of 
the cost of operating trains which may be affected by such 
changes of location as an engineer is able to make. There are 
some items of expense with which the engineer has not the 
slightest concern — nor will they be altered by any change in 
alignment or constructive detail which he may make. In the 
following discussion such items will be passed over with a brief 
discussion of the sub-items included. 

Maintenance of way. 

384. Item I. Repairs of Roadway. The item of repairs 
of roadway is very large — about half of the total cost of main- 
tenance of way and structures. It includes the cost of frogs, 
switches, switch-stands, and interlocking signals. The dis- 
tribution and laying of ties and rails, ballasting and tamping 

{turn to page 464) 



460 



RAILROAD CONSTRUCTION. 



§381. 



TABLE XX. SUMMARY SHOWING CLASSIFICATION OF OPERATING 
CENTAGE OF EACH CLASS TO TOTAL, CLASSIFIED 





Item. 


Amount. 


Per cent. 


No. 


1906. 


1906.1 


1905.2 




Maintenance of way and 
Structures. 

Repairs of roadway 


$164,468,769 
21,962,249 
38,467,183 

33,846,281 

6,330,746 

35,325,172 

3,695,079 

2,717,385 

459,273 

3,938,667 


10.726 
1.432 
2.509 

2.207 

.413 

2.304 

.241 
.177 
030 
.257 


10 393 




Renewals of rails 


1 316 




Renewals of ties 


2 657 




Repairs and renewals of bridges and 
culverts 

Repairs and renewals of fences, road 
crossings, signs, and cattle guards . . . 

Repairs and renewals of buildings and 
fixtures 


2.319 

.446 

2 114 




Repairs and renewals of docks and 
wharves 


208 


8 
9 


Repairs and renewals of telegraph .... 
Stationery and printing 


.*171 
028 


10 


Other expenses 


132 




Total 






$311,210,804 


20.296 


19 784 









Maintenance of Equipment. 

Superintendence 

Repairs and renewals of locomotives . 

Repairs and renewals of pass, cars . . . 

Repairs and renewals of freight cars .. 

Repairs and renewals of work cars . . 

Repairs and renewals of marine equip- 
ment 

Repairs and renewals of shop machin- 
ery and tools 

Stationery and printing 

Other expenses 

Total 



$8,612,019 
123,893,482 

30,177,532 

138,141,925 

4,107,826 

3,552,558 

10,252,866 

721,291 

8,633,469 



$328,092,968 



.561 
8.080 
1.968 
9.009 

.268 

.232 

.668 
.047 
.563 



21.396 



Conducting Transportation. 

Superintendence 

Engine and roundhouse men 

Fuel for locomotives 

Water-supply for locomotive.s 

Oil, tallow and waste for locomotives 

Other suppUes for locomotives 

Train service 

Train supplies and expenses . . . . . , 



$27,235,858 


1.776 


142,230.807 


9.275 


170,499,133 


11.119 


9,964,616 


.650 


5,903,014 


.385 


3,827,547 


.250 


97,757,296 


6.375 


23,871,258 


1.557 



1 Based on $1,533,404,385, excluding $3,472,886 unclassified 

2 " " 1,387,043,027, " 3,559,125 

3 •' '* 1,336,476,325, " 2,419,928 
* •* •• 1,254,936,972, " 2,601,880 
6 " *• 1,114,266,600. *' 1,965,680 



§381. 



OPERATING EXPENSES. 



461 



EXPENSES FOR THE YEAR ENDING JUNE 30, 1906, AND PER- 
FOR THE YEARS ENDING JUNE 30, 1897, TO 1906. 



Per cent. 




















Normal 


No. 


1904.3 


1903.4 


1902.5 


1901.6 


1900.7 


1899.8 


1898.9 


1897.10 


average 
for 10 
years. 




10.348 
1.298 
2.519 


11.093 
1.386 
2.487 


11.331 
1.521 
2.838 


10.924 
1.676 
3.140 


10.995 
1.138 
3.036 


10.720 
1.322 
2.901 


10.643 
1.391 
3.232 


10.644 
1.546 
3.357 


10.782 
1.403 
2.868 


1 
2 
3 


2.228 


2.461 


2.593 


2.730 


2.703 


2.374 


2.512 


2.472 


2.460 


4 


.437 


.527 


.625 


.598 


.616 


.487 


.537 


.509 


.519 


5 


2.147 


2.590 


2.562 


2.417 


2.466 


2.181 


1.957 


1.745 


2.248 


6 


.209 
.179 
.029 
.125 


.235 
.165 
.032 
.209 


.220 
.173 
.031 
.361 


.283 
.158 
.029 
.317 


.308 
.153 
.030 
.352 


.254 
.142 
.026 
.446 


.245 
.137 
.025 
.349 


.231 
.126 
.024 
.318 

20.972 


.243 
.158 
.028 
.287 


7 
8 
9 

10 


19.519 


21.185 


22.255 


22.272 


21.797 


20.853 

1 


21.028 


20.996 





.567 


.559 


.601 


.599 


.597 


.632 


.656 


.667 


.600 


11 


7.904 


7.408 


7.246 


6.695 


6.730 


6.208 


5.887 


5.663 


7.011 


12 


1.951 


2.044 


2.157 


2.277 


2.263 


2.164 


2.188 


2.265 


2.125 


13 


7.777 


7.442 


7.432 


7.436 


7.687 


7.038 


7.210 


6.376 


7.561 


14 


.231 


.242 


.245 


.233 


.252 


.210 


.159 


.140 


.222 


15 


.154 


.177 


.215 


.234 


.251 


.247 


.242 


.215 


.216 


16 


.704 


.696 


.643 


.605 


.604 


.512 


.486 


.478 


.606 


17 


.042 


.046 


.044 


.043 


.043 


.040 


.038 


.039 


.042 


18 


.637 


.519 


.544 


.507 


.502 


.544 


.493 


.509 


.542 


19 


19.967 


19.133 


19.127 


18.629 


18.929 


17.595 


17.359 


16.352 


18.925 





1.779 


1.742 


1.711 


1.726 


1.831 


1.767 


1.744 


1.845 


1.772 


20 


9.550 


9.562 


9.401 


9.340 


9.476 


9.690 


9.645 


9.922 


9.527 


21 


12.128 


11.675 


10.776 


10.602 


9.809 


9.478 


9.457 


9.392 


10.571 


22 


.659 


.614 


.623 


.612 


.599 


.619 


.646 


.677 


.636 


23 


.397 


.389 


.386 


.361 


.365 


.359 


.355 


.374 


.374 


24 


.248 


.232 


.218 


.206 


.188 


.177 


.156 


.160 


.207 


25 


6.735 


6.677 


6.737 


7.011 


7.244 


7.583 


7.660 


7.589 


7.015 


26 


1.581 


1.552 


1.500 


1.471 


1.467 


1.527 


1.525 


1.508 


1.527 


27 



« Based on $989,654,973. excluding $40,742,297 unclassified 

7 " " 923,432,555, " 37,995,956 

8 " •* 814,389,799, *' 42,579,200 

9 •• " 766,332,900, " 51,640,376 
10 •• •• 692,491,637, " 60,033,127 



462 



RAILROAD CONSTRUCTION. 



§381. 



TABLE XX- 



No. 



28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 



Item. 



Conducting Transportation — 
Continued 

S\\'itchman, flagmen, and watchmen . 

Telegraph expenses 

Station service 

Station supplies 

Switching charges, balance 

Car mileage, balance 

Hire of equipment, balance 

Loss and damage 

Injuries to persons 

Clearing wrecks 

Operating marine equipment 

Advertising 

Outside agencies 

Commissions 

Stockyards and elevators 

Rents for tracks, yards, and terminals. 
Rents for buildings and other property 

Stationery and printing 

Other expenses 

Total 



Amount. 



1906. 



$66,805,942 

26,853,012 

96,710,193 

9,362,704 

4,490,989 

18,885,086 

3,082,822 

21,086,219 

17,466,864 

4,601,240 

10,502,581 

6,467,954 

20,731.859 

267,394 

849,201 

26,848,580 

4,963,862 

9,639,066 

3,763,815 



Per cent. 



1906. 



$834,668,912 



4.357 

1.751 

6,307 

.611 

.293 

1.231 

.201 

1.375 

1.139 

.300 

.685 

.422 

1.352 

.017 

.055 

1.751 

.324 

.629 

.245 



54.432 



47 
48 
49 
50 
51 
52 

53 



General Expenses. 

Salaries of genersil officers 

Salaries of clerks and attendants . . . . 
General office expenses and supplies. . 

Insurance 

Law expenses 

Stationery and printing (general 

offices) 

Other expenses 



$12,660,837 

21,042,006 

4,028,647 

7,382,113 

6,938,807 

2,783,392 
4,595,899 



.826 
1.372 
.263 
.481 
.452 

.182 
.300 



Total . 



$59,431,701 3.876 



54 
55 
56 
57 



Recapitulation of Expenses. I 

Maintenance of wav and structures. . . ' $311,210,804 

Maintenance of equipment 1 328.092,968 

Conducting transportation ! 834.668.912 

General expenses i 59,431.701 

Grand total i $1,533,404,385 



20.296 19.784 

21.396 20.765 

54.432> 55.486 

3.876i 3.965 

I 

lOO.OOO'lOO.OOO 



§381. 

Continued. 



OPERATING EXPENSES. 



463 



Per Cent. 




















Normal 


No. 


1904. 


1903. 


1902. 


1901. 


1900. 


1899. 


1898. 


1897. 


average 
for 10 
years. 




4.386 


4.313 


3.984 


3.848 


3.944 


4.149 


4.080 


4.171 


4.157 


28 


1.788 


1.754 


1.784 


1.785 


1.812 


1.906 


1.907 


2.000 


1.828 


29 


6.605 


6.664 


6.832 


6.947 


7.103 


7.510 


7.758 


8.002 


7.017 


30 


.686 


.667 


.676 


.672 


.679 


.696 


.692 


.751 


.678 


31 


.280 


.244 


.272 


.319 


.340 


.356 


.331 


.327 


.307 


32 


1.358 


1400 


1.480 


1.618 


1.800 


2.010 


2.107 


2.203 


1.657 


33 


.195 


.214 


.180 


.161 


.223 


.368 


.342 


.249 


.235 


34 


1.279 


1.094 


.990 


.819 


.764 


.734 


.706 


.692 


.988 


35 


1.196 


1.120 


1.048 


.911 


.910 


.874 


.884 


.874 


1.011 


36 


.275 


.284 


.221 


.189 


.173 


.147 


.130 


.123 


.210 


37 


.696 


.745 


.721 


.862 


.866 


.868 


.958 


.934 


.805 


38 


.418 


.428 


429 


. .428 


.432 


.438 


.417 


.428 


.427 


39 


1.411 


1.449 


1.579 


1.615 


1.519 


1.781 


1.762 


1.727 


1.561 


40 


.022 


.044 


.077 


.089 


.151 


.194 


.181 


.158 


.095 


41 


.060 


.057 


.069 


.075 


.060 


.115 


.155 


.140 


.084 


42 


1.563 


1.544 


1.519 


1.724 


1.728 


1.901 


1.914 


1.953 


1.732 


43 


.382 


.411 


.440 


.440 


.464 


.487 


.475 


.501 


.427 44 


.640 


642 


.622 


.638 


.653 


.627 


.583 


.633 


.630: 45 


.353 


.376 
55.893 


.416 


.510 


.579 


.670 


.624 
57.194 


.587 
57.920 


.468 


46 


56.670 


54.671 


54.979 


55.179 


57.031 


55.946 





.841 


.823 


.925 


.984 


1.041 


1.171 


1.165 


1.235 


.985 


47 


1.313 


1.254 


1.244 


1.262 


1.269 


1.334 


1.334 


1.368 


1.309 


48 


.230 


.234 


.249 


.257 


.262 


.292 


.285 


.301 


.262 


49 


.471 


.432 


.412 


.384 


.349 


.372 


.395 


.436 


.423 


50 


.513 


.541 


.558 


.625 


.571 


.710 


.655 


.791 


.593 


51 


.170 


.175 


.168 


.161 


.166 


.171 


.176 


.163 


.171 


52 


.306 


.330 
3.789 


.391 
3.947 


.447 
4.120 


.437 


.471 


.409 


.462 


.390 


53 


3.844 


4.095 


4.521 


4.419 


4.756 


4.133 





19.519 

19.967 

56.670 

3.844 


21.185 

19.133 

55.893 

3.789 


22.255 

19.127 

54.671 

3.947 


22.272 

18.629 

54.979 

4.120 

100.000 


21.797 

18.929 

55.179 

4.095 

100.000 


20.853 

17.595 

57.031 

4.521 


21.028 

17.359 

57.194 

4.419 


20.972 

16.352 

57.920 

4.756 

100.000 


20.996 

18.925 

55.946 

4.133 


100.000 


100.000 


100.000 


100.000 


100.000 


100.000 



464 RAILROAD CONSTRUCTION. § 384. 

track, ditching, weeding, widening and protecting banks, the 
maintenance of snow-fences, dikes, and retaining walls, are also 
included. In short, any expense of maintaining the roadbed 
in condition which cannot be definitely assigned to one of the 
next few items will generally belong to this item — except per- 
haps those of item 10 {q.v.). The larger part of such items of 
expense is labor, and the variations will largely depend on the 
fluctuations in the wages of trackmen. About 1880 they had 
dropped to what Wellington considered to be a permanent 
average of $1.25 per day. In 1893 it had dropped to $1.22; 
then in 1897 and 1898 to $1.16. Since 1898 the average has 
steadily risen to $1.36 in 1906. 

In 1899 the average cost of this item per mile of main track 
was about $480, but this figure, after all, is of but little value 
because, for the reason already given in general in § 381, it will 
be found that the cost for any road varies almost exactly as 
the train-mileage and will average very closely to lie. per train- 
mile, whether the traffic be heavy or light. 

385. Item 2. Renewal of Rails. This item may be con- 
sidered as having been withdrawn from the previous item 
simply because it is one of the largest of the single items and 
because its cost is very readily determined. It includes the 
cost of rails, their inspection, and their delivery (but not 
their distribution). The item shows a large percentage of vari- 
tion. It increased from 1.138% in 1900 to 1.676% in 1901, 
an increase of over 47%. Such fluctuations are due first to 
that considerable fluctuation in the price of rails which rail- 
roads can hardly expect to escape, and secondly to variations 
in the standard of maintenance caused first by hard times, 
which are then followed by unusual expenditures in good times, 
or by the expenditures absolutely essential to restore the track 
to its former condition. The item includes all rails wherever 
used, whether on main track, siding, repair track, gravel track, 
on wharves or coal-docks, and even includes guard-rails. But 
it does not include any rail attachments such as joints, frogs, 
switches, etc. The rate of rail wear under various conditions 
has already been discussed in Chapter IX. 



§ 388. OPERATING EXPENSES. 465 

386. Item 3. Renewal of Ties. As with the previous 
item, this item is simply a detachment from the general item, 
repairs of roadway. As wuth rails, the cost of laying and dis- 
tributing the ties is not included, but the cost of tie-plates and 
tie-plugs, also chemical treatment for preservation, if such is 
used, is included in this item. While the cost will vary con- 
siderably between different roads on account of first cost, kind 
of w^ood, climate, etc., the item for any one road for a period of 
years cannot vary greatly, unless there is a marked change in 
the standard of maintenance. The actual cost of such work 
has already been discussed in Chapter VIII. 

387. Item 4. Repairs and Renewals of Bridges and 

CULVERTS; This item includes not only the maintenance cost 
of all bridges, trestles, viaducts, and culverts, but of all piers, 
abutments, riprapping, etc., necessary to maintain them, and 
even the cost of operating drawbridges. The locating engi- 
neer is not concerned with this item, except as he may con- 
sider that some distance which is to be added (or cut out) has 
the average number of culverts and bridges. With culverts 
and small bridges there would be little or no error in such an 
assumption, but if there were any large bridges on the portion 
of track under discussion, they w^ould need special consideration. 

388. Items 5 to 10. Repairs and Renewals of Fences, 
Road Crossings, and Cattle-guards— Of Buildings and 
Fixtures— Of Docks and Wharves— Of Telegraph Plant; 
Stationery and Printing; and "Other Expenses." These 
items in the aggregate amount to but 3.5% of the average cost 
per train-mile. The fluctuations have so small an effect on the 
average cost per train-mile that they may be neglected. In 
item 5 are included not only those things which are specifically 
mentioned, but also those structures which in general are not 
directly affected by the running of trains. For example, "road 
crossings'' include not only the maintenance of highw^ay cross- 
ings at grade, but also overhead highway crossings and what- 
ever a railroad may have to pay for the maintenance of a bridge 
by which another railroad crosses it. On the other hand, the 
maintenance of a bridge by which a railroad crosses another 
road (highway or railroad) is charged to bridges. The effect 
(if any) of these items on any changes in construction which 
an engineer may make will be specifically discussed in the suc- 
ceeding chapters. 



466 RAILROAD CONSTRUCTION. § 3S9. 



MAINTENANCE OF EQUIPMENT. 

389. Item II. Superintendence i This item includes those 
fixed charges in superintendence which do not fluctuate with 
small variations in business done. It includes the salaries 
of superintendent of motive power, master mechanic, master 
car-builder, foremen, etc., but does not include that of road 
foremen of engines nor enginemen. In a general way the item 
is proportional to the general scale of business of the road, but 
does not fluctuate with it. 

390. Item 12. Repairs and Renewals of LocoMOTivESi 

This item must be studied by the locating engineer in order to 
determine the effect on locomotive repairs and renewals of an ad- 
dition to distance (considered in Chapter XXI), the effect (chiefly 
in wheel wear) of a reduction in curvature (considered in Chap- 
ter XXII), or the effect of grade (considered in Chapter XXIII). 
In studying the effect of grade, the policy of adopting heavier 
locomotives and the effect of this on this item must also be 
considered. This item includes the expenses of work whose 
effect is supposed to last for an indefinite period. It does not 
include the expense of cleaning out boilers, packing cylinders, 
etc., which occurs regularly and which is charged to item 21, 
round-house men. It does include all current repairs, general 
overhauling, and even the replacement of old and worn-out 
locomotives by new ones to the extent of keeping up the original 
standard and number. Of course additions beyond this must 
be considered as so much increase in the original capital invest- 
ment. As a locomotive becomes older the annual repair charge 
becomes a larger percentage on the first cost, and it may be- 
come as much as one fourth and even one third of the first cost. 
When a locomotive is in this condition it is usually consigned to 
the scrap-pile; the annual cost for maintenance becomes too 
large an item for its annual mileage. The effect on expenses 
of increasing the weight of engines is too comphcated a prob- 
lem to admit of precise solution, but certain elements of it may 
be readily computed. While the cost of repairs is greater for 
the hea^der engines, the increase is only about one half as fast 
as the increase in weight — some of the sub-items not being 
increased at all. 



§ 394. OPERATING EXPENSES. 467 

391. Items 13, 14, 15. Repairs and Renewals of Pas- 
senger Cars, of Freight Cars, and of Work Cars* As 

with engine repairs, the item excludes consumable supplies (oil, 
waste, illuminating oil or gas, ice, etc.), but includes in general 
all items necessary to maintain the cars up to the full standard 
of condition and number, and even to replace old worn-out 
cars by new. When, as is frequently the case with both cars 
and locomotives, the new rolling stock is larger, better, and of a 
higher standard than that which is replaced, the difference in 
cost should be added to capital investment. The chief con- 
cern of the locating engineer regarding this item is the effect 
on car repairs of additional distance, of variations in curva- 
ture (affecting wheel wear chiefly), and of grade (affecting the 
draft-gear and general wear and tear). These items will be 
considered under their proper heads in the following chapters. 

392. Items 16, 17, 18, and 19. REPAIRS AND RENEWALS 
OF Marine Equipment — Of Shop Machinery and Tools; 
Stationery and Printing; Other Expenses. The location 
of the road along the line has no connection with the main- 
tenance of marine equipment. The maintenance of shop 
machinery and tools can only be affected as the work of repairs 
of rolling stock fluctuates, and of course in a much smaller 
ratio. No change which an engineer can effect will have any 
appreciable influence on this item. 

The other items are too small and have too little connection 
with location to be here discussed except as it may be considered 
that they vary with train mileage, which an engineer may 
influence (see Chapter XXIII, Grades). 



CONDUCTING TRANSPORTATION. 

393. Item 20. Superintendence. As with item 11, this 
item is not subject to minor fluctuations in business, but only 
varies with changes in the general scale of the business of the 
road. 

394. Item 21. Engine and Round-house Men. This item 
includes the wages of engineers, firemen, and also all men em- 
ployed around the engine-houses except those who are making 
such repairs as should be charged to maintenance of equipment 
(item 12). The item is a large one, but is only affected by one 
class of change of location — a difference in length of line. The 



468 RAILROAD CONSTRUCTION. § 394. 

wages of the round-house men constitute but a small percentage 
of this item, and the wages of the enginemen vary almost directly 
as the mileage. On very short roads, where the number of 
round trips which may properly constitute a da3^'s work is 
definitely limited and on which there is but little night or Sunday 
work, the wages may be practically by the day, and a variation 
in length of several hundred feet or even a few miles in the 
length of the road may make practically no difference in the 
w^ages paid. But on the larger roads, operated by divisions, 
on which (especial!}^ in freight work) there is no distinction of 
day or night, week day or Sunday, the varying length of divisions 
is equalized by calhng them IJ or IJ runs, a ''run" usually 
being considered as about 100 miles. The enginemen are then 
paid according to the number of runs made per month. The 
effect on this item of variations in distance is discussed more 
fully in Chapter XXI. 

395. Item 22. Fuel for Locomotives. The item includes 
the entire cost of the fuel until it is placed in the engine-tender. 
The cost therefore includes not only the first cost at the point 
of dehvery to the road, but also the expense of hauling it over 
the road from the point of delivery to the various coaling stations 
and the cost of operating the coal-pockets from which it is 
loaded on to the tenders. Although the cost is fairly regular 
for any one road, it is exceedingly variable for different roads. 
Roads running through the coal regions can often obtain their 
coal for eighty or ninety cents per ton. Other roads far re- 
moved from the coal-mines have been compelled to pay six dollars 
per ton. In the three succeeding chapters there will be con- 
sidered in detail the effect on fuel consumption of variations 
in location. It will be shown that fuel consumption is quite 
largely independent of distance and the number of cars hauled. 

396. Items 23, 24, and 25. WATER-SUPPLY ; OIL, TALLOW, 
AND Waste ; Other Supplies for Locomotives; The cost of 

the water-supply is quite largely a fixed charge except where 
it is supplied by municipalities at meter rates. The consump- 
tion of all these supplies will vary nearly as the engine-mileage. 

397. Item 26. Train Service. This item is one of the 
largest single items and includes in general the wages of all 
the train-hands except the enginemen. As with enginemen, 
they are paid according to the number of runs. The item is 
therefore of importance to the locating engineer from the one 



§ 400. OPERATING EXPENSES. 469 

standpoint of distance, and even then only when the variation 
in distance which is considered will afTect the classification of 
the run and therefore the rate of pay for that run. 

398. Item 27. Train Supplies and Expenses. These in- 
clude the large list of consumable supplies such as lubricating 
oil, illuminating oil or gas, ice, fuel for heating, cleaning materials, 
etc., which are used on the cars, and not on the locomotives. 
The consumption of some of these articles is chiefly a matter 
of time ; — in other cases it is a function of the mileage. 

399. Items 28, 29, 30, and 31. SWITCHMEN, FLAGMEN, AND 

Watchmen; Telegraph Expenses; Station Service; and 
Station Supplies. These items will be proportional to the 
general scale of business of the road, but are independent of small 
fluctuations in business. The main items are ob\'ious from the 
titles. Many sub-items, which are very small or are of occasional 
or accidental occurrence, are also included under these items for 
lack of a better classification. 

400. Items 32,33, and 34. SWITCHING CHARGES— BALANCE ; 

Car Mileage— Balance; Hire of Equipment. The first of 
these is a charge paid by a road to other corporations for 
switching done for the road. The locating engineer is not 
concerned with this item. 

Car Mileage. This is a charge paid by a road for the use 
of the cars (chiefly freight cars) of another road. To save the 
rehandling of freight at junctions the policy of running freight 
cars on to foreign roads is very extensively adopted. Since 
the foreign road receives (ultimately) its mileage proportion 
of the freight charge, it justly pays the home road a rate which 
is supposed to represent the value of the use of a freight car 
for so many miles. The foreign road then loads up the freight 
car with freight consigned to some point on the home road and 
sends it back, again paying mileage for the distance traveled 
on the foreign road, a proportional freight charge having been 
received for that serWce. By a clearing-house arrangement 
the various roads settle their debit and credit accounts Tvath 
each other by the payment of a balance. Such is the simple 
theory. In practice the cars are not sent back to the home 
road at once, but wander off according to the local demand. 
As long as strict account is kept of the movements of every 
car and the home road is paid a charge which really covers 
the value of such service, no harm is done the home road except 



470 RAILROAD CONSTRUCTION. § 400. 

that sometimes, when business has suddenly increased, the 
home road cannot get enough cars to handle its business. The 
value of a car is then abnormally above its ordinary value 
and the home road suffers for lack of the rolhng stock which 
belongs to it. The charge being paid according to mileage, 
any variations of distance have a direct bearing on this item. 

Hire of Equipment. This may refer to locomotives or cars 
which are hired for a special service, or, on very poor roads, 
it may refer to equipment, which is hired rather than purchased. 
The locating engineer has no concern with this item. 

401. Items 35, 36, and 37. LOSS AND DAMAGE; INJURIES 
TO Persons; Clearing Wrecks. These expenses are fortuitous 
and bear no absolute relation to road-mileage or train-mileage. 
While they depend largely on the standards of discipline on 
the road, even the best of roads have to pay some small pro- 
portion of their earnings to these items. The possible relation 
between curvature and accidents is discussed in Chapter XXII, 
but otherwise the locating engineer has no concern \vdth these 
items. 

402. Items 38 to 53. All of the remaining items (for a list 
of which see Table XX) are of no concern to the locating engineer. 
They are either general expenses (such as taxes) or are special 
items (such as the operation of marine equipment) which will 
not be changed by variations in distance, curvature, or grades 
which a locating engineer may make. They will not therefore 
be further discussed. 



CHAPTER XXL 
DISTANCE. 

403. Relation of distance to rates and expenses. Rates 
are usually based on distance traveled, on the apparent 
hypotheses that each additional mile of distance adds its pro- 
portional amount not only to the service rendered but also to 
the expense of rendering it. Neither hypothesis is true. The 
value of the service of transporting a passenger or a ton of 
freight from A to 5 is a more or less uncertain gross amount 
depending on the necessities of the case and independent of 
the exact distance. Except for that very small part of passen- 
ger traffic which is undertaken for the mere pleasure of traveling, 
the general object to be attained in either passenger or freight 
traffic is the transportation from A to B, however it is attained. 
A mile greater distance does not improve the service rendered; 
in fact, it consumes valuable time of the passengers and perhaps 
deteriorates the freight. From the standpoint of service ren- 
dered, the railroad which adopts a more costly construction and 
thereby saves a mile or more in the route between two places 
is thereby fairly entitled to additional compensation rather 
than have it cut down as it would be by a strict mileage rate. 
The actual value of the service rendered may therefore vary 
from an insignificant amount which is less than any reasonable 
charge (which therefore discourages such traffic) and its value 
in cases of necessity — a value which can hardly be measured in 
money. If the passenger charge between New York and Phila- 
delphia were raised to $5, $10, or even $20, there would still be 
some passengers who w^ould pay it and go, because to them 
it would be worth $5, $10, or $20, or even more. Therefore, 
when they pay $2.50 they are not paying what the service is 
worth to them. The service rendered cannot therefore be 
made a measure of the charge, nor is the service rendered pro- 
portional to the miles of distance. 

The idea that the cost of transportation is proportional to 

471 



472 RAILROAD CONSTRUCTION. § 403. 

the distance is much more prevalent and is in some respects 
more justifiable, but it is still far from true. This is especially 
true of passenger service. The extra cost of transporting a single 
passenger is but little more than the cost of printing his ticket. 
Once abroad the train, it makes but little difference to the rail- 
road whether he travels one mile or a hundred. Of course there 
are certain very large expenses due to the passenger traffic 
which must be paid for by a tariff which is rightfully demanded, 
but such expenses have but little relation to the cost of an 
additional mile or so of distance inserted between stations. 
The same is true to a slightly less degree of the freight traffic. 
As sho^\Tl later, the items of expense in the total cost of a train- 
mile, which are directly affected by a small increase in distance, 
are but a small proportion of the total cost. 

404. The conditions other than distance that affect the cost; 
reasons why rates are usually based on distance. Curvature 
and minor grades have a considerable influence on the cost of 
transportation, as will be sho\\Ti in detail in succeeding chap- 
ters, but they are never considered in making rates. Ruling 
grades have a very large influence on the cost, but they are like- 
wise disregarded in making rates. An accurate measure of 
the effect of these elements is difficult and complicated and 
would not be appreciated by the general public. Mere dis- 
tance is easily calculated; the public is satisfied with such 
a method of calculation; and the railroads therefore adopt a 
tariff which pays expenses and profits even though the charges 
are not in accordance with the expenses or the service rendered. 

An addition to the length of the line may (and generally does) 
involve curvature and grade as well as added distance. In 
this chapter is considered merely the effect of the added dis- 
tance. The effect of grade and curvature must be considered 
separately, according to the methods outlined in succeeding 
chapters. The additional length considered is likeT\dse assumed 
not to affect the business done nor the number of stations, but 
that it is a mere addition to length of track. 

EFFECT OF DISTANCE ON OPERATING EXPENSES. 

405. Effect of slight changes in distance on maintenance of 
way. With a few unimportant exceptions all the items of 
expense under maintenance of way and structures (see § 407) 



§ 406. DISTANCE. 473 

will be increased directly as any increase in distance. . This 
must certainly be true for items 1, 2, 3, and 5, which alone 
comprise about three fourths of the total expense for mainte- 
nance of way. If we assume that the proposed change of length 
involves no difference in the number of bridges, culverts, build- 
ings, and fixtures,* docks and wharves, we may consider items 
4, 6, and 7 to be unaffected. This will generally be true for 
small changes in length, measured in feet. For larger differ- 
ences, measured in miles, item 4 will vary nearly as the 
distance. The same may be said of items 9 and 10. The tost 
of maintaining the telegraph line will probably be increased 
about 50% of tke unit cost. The effect of changes in distance 
on these various items of maintenance of way (as well as the 
other items of expense of a train-mile) will be tabulated in § 408. 

406. Effect on maintenance of equipment. The relation 
between an increase in length of line and the expenses of items 
11, 15, 17, 18, and 19 are quite indefinite. In some respects 
they would be unaffected by slight changes of distance. From 
other points of view there is no reason why the expenses should 
not be considered proportionate to the distance. For exam- 
ple, the added track will probably require as much work from 
the construction train as any other part of the road and is 
therefore responsible for as much of the ^'repairs and renewals 
of Y/ork-cars" — item 15. Fortunately all of these items are so 
small, even in the aggregate, that little error will be involved 
by either decision. It will therefore be assumed that these 
items are affected 100% for large additions in distance and but 
50% for small additions. 

Item 16 is evidently unaffected. 

Item 12. Locomotives deteriorate (1) with age; (2) by 
expansion and contraction, especially of the fire-boxes, when 
fires are drawn and relighted; (3) on account of the strains due 
to stopping and starting; (4) the strains and wear of wheels due 
to curved track; (5) the additional stresses due to grade and 
change of grade: and (6) on account of the Avork of pulling 
on a straight level track. Observe that the first five causes 
have no direct relation to an addition of mere distance (the 
possible curvature or grade incident to the additional distance 
being a separate matter). How much of the total deteriora- 
tion is due to the last cause? Wellington attacks this problem 
as follows: the records of engine-repair shops readily furnish 



474 RAILROAD CONSTRUCTION. § 406. 

the proportionate cost of the repairs of boiler, running-gear, 
etc. An estimate is then made of the effect of each cause on 
each item. For example, the boiler is responsible for 20% of 
the repairs and renewals. Of this 7% (say one third) is assigned 
to "terminal service, getting up steam, making up trains," 
4% to curvature and grades, 2% to ''stopT[Ding and starting 
at way stations," and the other 7% to '^ distance on tangent 
between stations." The other items are treated similarly. 
Wellington says, "As this [subdi\4sion of expenses] has been 
done with great care to get the best attainable authorit}^ for 
each (which it would occupy too much space to give in detail), 
the margin for possible error is not great enough to be of mo- 
ment, although no absolute exactness can be claimed for it." 
His final estimate is that distance is responsible for 42% of 
the total cost of repairs and renewals. This value will there- 
fore be used for all additional distances, groat or small. 

Items 13 and 14. The causes of deterioration of both passenger 
and freight cars may be classified exactly as above — omitting 
merely cause 2 — the expansion and contraction due to firing. 
Considering that a large part of the repairs of freight cars is 
due to the draft-gear and brakes, which are affected chiefly 
by the heavy strains due to stopping and starting and to grades, 
while the repairs of wheels are largely due to the wear of wheels 
on curves, it is not surprising that he allows only 36% of the 
cost of repairs and renewals of freight cars to be due to straight 
distance. He made no direct estimate for passenger-cars, but 
points out the fact that the maintenance of the seats, furniture, 
and ornamentation make up much more than half the cost 
of passenger-car repairs. A large part of such deterioration 
is due to age and the weather, although that of the seats is 
largely a function of passenger wear and therefore of distance 
traveled. Although the items of deterioration in passenger 
cars is very different from those of freight cars yet if a similar 
calculation is made for passenger cars it will be found that the 
final figure is substantially the same as for freight cars and will 
here be so regarded. 

407. Effect on conducting transportation. Item 20. This is 
CA-idently unaffected by small or even considerable additions 
to distance. 

Item 21. Theoretically, train wages should var}^ as mileage. 
On the larger roads, where, especially in the freight service, 



§ 407. DISTANCE. 475 

there is little or no distinction of day or night, week-day or Sun- 
day, it is practically impossible to hire the trainmen to work 
between certain definite hours of the day and pay them accord- 
ingly, as is done with factory employees. As explained in Chap- 
ter XX, § 394, the system usually adopted of paying trainmen is 
such that small changes of distance (measured in feet) would 
not affect train wages. The wages of round-house men would 
not be affected under any conditions, and those of the enginemen 
and of the trainmen (item 26) would not generally be affected 
unless the change of distance is very great — perhaps ten miles. 
Since items 21 and 26 are both very large, it \\dll not do to 
ignore this item or to average it. The pay of round-house men 
is about 9% of item 21. We may therefore say that if the 
change in distance is so great that trains wages will be affected, 
item 21 will be affected 91% and item 26 will be affected 100%. 
For shorter changes of distance they will be unaffected. 

Item 22. A surprisingly large percentage of the fuel con- 
sumed is not utilized in drawing a train along the road. Part 
of this loss is due to firing up, part is wasted when the engine 
is standing still, which is a large part of the total time. The 
policy of banking fires instead of drawling them reduces the 
injury resulting from great fluctuations in temperature, but 
the total coal consumed is about the same and we may there- 
fore consider that almost a fireboxful of coal is wasted whether 
the fires are banked or drawn. The amount thus wasted (or at 
least not utilized in direct hauling) has been estimated at 5 to 
10% of the whole consumption. Experiments* have show^n that 
an engine standing idle in a yard, protected from wind, well 
jacketed, etc., will require from 25 to 32 lbs. of coal per hour 
simply to keep up steam. It has been found that the fastest 
express trains will lose one fourth of their total time between 
termini in stops, and freight trains on a single-track road will 
generally spend four hours per day on sidings. The waste of 
coal from this cause is estimated at 3 to 6% of the total con- 
sumption. The energy consumed in stopping and starting is 
very great. A train running 30 miles per hour has enough 
kinetic energy to move it on a level straight track more than 
two miles. Every time a train running at 30 miles per hour 
is stopped, enough energy is consumed by the brakes to run 

* Wellington, Economic Theory, p. 207. 



476 RAILROAD CONSTRUCTION. § 407. 

it from one to two miles. When starting, it will require an 
equal amount of work to restore that velocity, in addition to 
the ordinary resistances. It has been shown that on the Man- 
hattan Elevated Railroad, where stops wdll average every three 
Eighths of a mile, this cause alone w^ill account for the consump- 
tion of nearly three fourths of the fuel. Of course on ordinary 
railroads the proportion is not nearly so great, but it is probably 
as much as 10 to 20% as an average figure. For a through 
express train making but few stops the figure would be small, 
except for the effect of ^' slow-doA\'ns." For suburban trains 
the proportion would be abnormally high. The fuel required 
to overcome the added resistances due to curvature and grade 
are of course exceedingh^ variable, depending on the particular 
alignment of the road considered. An approach to the truth 
may be made by considering the average curvature per mile 
for the roads of the United States and the average grades, 
and computing, by the methods given in subsequent chapters, 
the extra fuel consumed on account of such average conditions, 
and these items will apparently be responsible for 4% due to 
curvature and about 25% due to grades. Summarizing the 
above w^e have: 

Firing 5 to 10% 

Wasted while still 3 ' ' 6%, 

Stopping and starting 10 '' 20% 

Average curvature 4 " 4% 

Average grade 25 ' ' 25% 

47 65 
Direct hauling 53 ' ' 35 Average, 44% 

100 100 

This shows that the addition of mere straight level distance 
would not increase the consumption of fuel more than 44% of 
the average consumption per mile. 

Items 23, 24, and 25. If water is paid for by meter, the cost 
is strictly according to consumption, which would vary almost 
according to the number of engine-miles. When supplied 
from the company^s own plant, as is usually the case, a slight 
increase will not appreciably affect the cost. Nothing is wasted 
during firing or while the engine is still. The use is therefore 
more nearly as the mileage, and the cost for an additional mile 



§ 408. DISTANCE. 477 

may be considered as 50% of its arverage cost per train-mile. 
Items 24 and 25 will be considered similarly. Fortunately 
these items, whose variation with additional distance is some- 
what obscure and variable, only aggregate a little over 1% of 
the cost of a train-mile and therefore a considerable percentage 
of error is of little or no importance. 

Item 26. (See comments on item 21.) 

Item 27. This item, as well as many other small items that 
follow, will be irregularly affected b}^ a small increase in distance. 
It would appear equally wrong to say that they would be un- 
affected or to say that they will vary directly as the mileage. 
50% w^ill be allowed. 

Item 28. The necessity for flagmen and watchmen varies 
in general as the mileage. An addition in distance is less apt 
to increase the number of switchmen. 50% of this item will 
be added for great distances and 25% for small distances. 

Item 29. Telegraph expenses include the wa^es of operators 
(unaffected), and the special expenses due to offices and tele- 
graph stations and to operating the line — the maintenance of 
the line being charged to item 8. This item will be so little 
affected by additional distance that nothing will be allowed. 
Items 30, 31, 32, and 34 are unaffected. Items 33, 35, 36, and 
37 are affected 100%. Items 38 to 46 are unaffected. 

The *' general expenses" (items 46 to 53) will be unaffected. 

408. Estimate of total effect on expenses of small changes 
in distance (measured in feet) ; estimate for distances measured 
in miles. According to the accompanying compilation the cost 
of operating additional distance will be about 34% of the 
average cost per train-mile when the additional distance is small, 
but will be about 54.5% if the additional distance is several 
miles. The figures may also be considered as the saving in th« 
operating expenses resulting from a shortening of the line. 

The average cost of a train-mile has been steadily rising foi 
many years past — see § 380. It seems impossible that the 
rise can continue indefinitely. On the basis of $1.35 per train- 
mile the above figures become 45.9 and 73.6 cents per train- 
mile respectively. Some trains run 365 days per year, others 
but 313. The tendency is toward the larger figure and it will 
therefore be used in these calculations. The added cost per 
daily train per year for each foot of distance is 

45.9X365X2 ^ ^, 
—^280 =^^^^- 



478 



Railroad coNSTRUCTiGisf. 



§408. 



When the distance is measured by miles the added cost per 
daily train per year for each mile of distance is: 

$0,736X365X2 = $537. 



TABLE XXI. — EFFECT ON OPERATING EXPENSES OF GREAT 
(and small) changes IN DISTANCE. 





t 


Per cent 
affected. 


Cost per mile. 


i 


1 


Per cent 
affected. 


Cost per mile. 


Q 


0) 

c3 










M 


> 








"A 


















a 


o 






Great. 


Small. 


2 
1 .2 


6 


c3 


1 


Great. 


Small. 




iz; 


O 


m 






i 


"A 


o 


m 


















j 


23.087 






13.93 


5.26 


*i 


10.782 


100 


100 


10.78 


10.78 


i 26 


7.015 


lOO 


' ' d 


7.01 





2 


1.103 


100 


100 


1.40 


1.40 


27 


1 .527 


50 


50 


.76 


.76 


3 


2.868 


100 


100 


2.87 


2.87 


1 28 


4.157 


50 


25 


2.08 


1.04 


4 


2.480 


100 





2.46 





1 29 


1.828 














6 


.519 


100 


100 


.52 


.52 


\ 30 


7.017 














6 


2.248 














31 


.678 














7 


.243 














32 


.307 














8 


.158 


50 


50 


.08 


.08 


' 33 


1.657 


100 


100 


1.66 


1.66 


9 


.028 


100 





.03 





34 


.2351 











10 


.287 


100 





.29 





35 


.988 100 


100 


.99 


.99 














, 36 
37 

1 38 
39 


1.011 100 
.210 100 
.805 1 
.427 


100 
100 


1.01 
.21 


1.01 
.21 




20.996 






18.43 


15.65 










11 


.600 


100 


50 


.60 


.30 








12 


7.011 


42 


42 


2.95 


2.95 


! 40 


1.561 


1 








13 


2.125 


36 


36 


.76 


.76 


41 


.095 










14 


7.561 


36 


36 


2.72 


2.72 


42 


.084 


\ 











15 


.222 


100 


50 


.22 


.11 


43 


1.732 










16 


.216 














44 


.427 










17 


.606 


100 


50 


.61 


.30 


45 


.630 










18 


.042 


100 


50 


.04 


.02 


46 


.468 










19 


.542 


100 


50 


.54 


.27 
















55.916 






27.65 


10.93 




18.925 






8.44 


7.43 










. . . . 




47 
48 












20 


1.772 






















21 


9.527 


91 





8.67 





49 












22 


10.571 


44 


44 


4.65 


4.65 


50 


f4.133 














23 


.636 


50 


50 


.32 


.32 


51 










24 


.374 


50 


50 


.19 


.19 


52 












25 


.207 


50 


50 


.10 


.10 


53 


100.000 












23.087 


....!.... 


13.93 


5.26 




.... 


54.52 


34.01 




i 


■■■•l"" 





* For the significance of the items, see Table XX. 

Light-traffic roads are more apt to run their trains on week 
days only, and a corresponding reduction should be made in 
these cases. 

Regarding the accuracy of the above computations, it should 
be noted that the must uncertain items are generally the smallest, 
and that even the largest variations that can reasonably be 



§ 410. * DISTANCE. 479 

made of the above figures will not very greatly alter the final 
result. A numerical illustration of the value of saving distance 
will be given later. 

EFFECT OF DISTANCE ON RECEIPTS. 

409. Classification of traffic. There are various methods 
of classifying traffic, according to the use it is intended to 
make of the classification. The method here adopted will have 
reference to its competitive or non-competitive character and 
also to the method of division of the receipts on through traffic. 
Traffic may be classified first as ^ through" and '^ocal" — 
through traffic being that traveling over two (or |'more) lines, 
no matter how short or non-competitive it may be; ^' local" 
traffic is that confined entirely to one road. A fivefold classifica- 
tion is however necessary — which is: 

A. Non-competitive local — on one road with no choice of 
routes. 

B. Non-competitive through — on two (or more) roads, but 
with no choice. 

C. Competitive local — a choice of two (or more) routes, but 
the entire haul may be made on the home road. 

D. Competitive through — direct competition between two 
or more routes each passing over two or more lines. 

E. Semi-competitive through — a non-competitive haul on the 
home road and a competitive haul on foreign roads. 

There are other possible combinations, but they all reduce to 
one of the above forms so far as their essential effect is concerned. 

410. Method of division of through rates between the 
roads run over. Through rates are divided between the 
roads run over in proportion to the mileage. There may 
be terminal charges and possibly other more or less arbitrary 
deductions to be taken from the total amount received, but 
when the final division is made the remainder is divided accord- 
ing to the mileage. On account of this method of division and 
also because non-competitive rates are always fixed according 
to the distance, there results the unusual feature that, unlike 
cur^^ature and grade, there is a compensating advantage in 
increased distance, which applies to all the above kinds of 
traffic except one (competitive local), and that the compensation 
is sometimes sufficient to make the added distance an actual 



480 RAILROAD CONSTRUCTION. § 411. 

source of profit. It has just been proved that the cost of hauhng 
a train an additional mile is only 34 to 54% of the average 
cost. Therefore in all non-competitive business (local or 
through) where the rate is according to the distance, there is 
an actual profit in all such added distance. In competitive 
local business, in which the rate is fixed by competition and 
has practically no relation to distance, any additional distance 
is dead loss. In competitive through business the profit or 
loss depends on the distances involved. This may best be 
demonstrated by examples. 

411. Effect of a change in the length of the home road on 
its receipts from through competitive traffic. Suppose the 
home road is 100 miles long and the foreign road is 150 miles 

long. Then the home road will receive -— -r — r-r^ =40% of the 

through rate. 

Suppose the home road is lengthened 5 miles; then it will 

105 
receive ^TTT^ — -^^=41.176% of the through rate. The traffic 
105 + 150 

being competitive, the rate will be a fixed quantity regardless 
of this change of distance. By the first plan the rate received 
is 0.4% per mile; adding 5 miles, the rate for the original 100 
miles may be considered the same as before; and that the addi- 
tional 5 miles receive 1 . 176%, or 0. 235% per mile. This is 59% 
of the original rate per mile, and since this is more than the 
cost per mile for the additional distance (see § 408), the added 
distance is e^ddently in this case sl source of distinct profit. 
On the other hand, if the line is shortened 5 miles, it may be 
similarly shown that not only are the receipts lessened, but 
that the saving in operating expenses by the shorter distance 
is less than the reduction in receipts. 

A second example will be considered to illustrate another 
phase. Suppose the home road is 200 miles long and the foreign 
road is 50 miles long. In this case the home road will receive 

onn ■ e^n =^^% ^^ ^^^ through rate. Suppose the home road is 

205 
lengthened 5 miles; then it will receive ^^rrr — r^ = 80.392% 

of the through rate. By the first plan the rate received is 
. 400% per mile ; adding 5 miles, there is a surplus of . 392, 
or 0.0784 per mile, which is but 19.6% of the original rate. 



§ 414. DISTANCE. 481 

At this rate the extra distance evidently is not profitable, al- 
though it is not a dead loss — there is some compensation. 

412. The most advantageous conditions for roads forming 
part of a through competitive route. From the above it may 
be seen that when a road is but a short link in a long com- 
petitive through route, an addition to its length will increase 
its receipts and increase them more than the addition to the 
operating expenses. 

As the proportionate length of the home road increases the 
less will this advantage become, until at some proportion an 
increase in distance w^ill just pay for itself. As the proportionate 
length grows greater the advantage becomes a disadvantage 
until, when the competitive haul is entirely on the home road, 
any increase in distance becomes a net loss without any com- 
pensation. It is therefore advantageous for a road to be a 
short link in a long competitive route; an increase in that link 
will be financially advantageous; if the total length is less than 
that of the competing line, the advantage is still greater, for 
then the rate received per mile will be greater. 

413. Effect of the variations in the length of haul and the 
classes of the business actually done. The above distances 
refer to particular lengths of haul and are not necessarily the 
total lengths of the road. Each station on the road has 
traffic relations with an indefinite number of traffic points 
all over the country. The traffic between each station on 
the road and any other station in the country between which 
traffic may pass therefore furnishes a new combination, the 
effect of which will be an element in the total effect of a 
change of distance. In consequence of this, any exact solution 
of such a problem becomes impracticable, but a sufficiently 
accurate solution for all practical purposes is frequently ob- 
tainable. For it frequently happens that the great bulk of a 
road's business is non-competitive, or, on the other hand, it 
may be competitive-through, and that the proportion of one 
or more definite kinds of traffic is so large as to overshadow 
the other miscellaneous traffic. In such cases an approximate 
but sufficiently accurate solution is possible. 

414. General conclusions regarding a change in distance. 
(a) In all non-competitive business (local and through) the 
added distance is actually profitable. Sometimes practically 



482 RAILROAD CONSTRUCTION § 414. 

all of the business of the road is non-competitive ; a considerable 
proportion of it is always non-competitive. 

(b) When the competitive local business is very large and the 
competitive through business has a very large average home 
haul compared with the foreign haul, the added distance is 
a source of loss. Such situations are unusual and are generally 
confined to trunk lines. 

(c) The above may be still further condensed to the general 
conclusion that there is always some compensation for the added 
cost of operating an added length of line and that it frequently 
is a source of actual profit. 

(d) There is, however, a limitation which should not be lost 
sight of. The above argument may be carried to the logical 
conclusion that, if added distance is profitable, the engineer 
should purposely lengthen the line. But added distance means 
added operating expenses. A sufficient tariff to meet these is a 
tax on the community — a tax which more or less discourages 
traffic. It is contrary to public policy to burden a community 
with an avoidable expense. But, on the other hand, a railroad 
is not a charitable organization, but a money-making enter- 
prise, and cannot be expected to unduly load up its first cost 
in order that subsequent operating expenses may be unduly 
cheapened and the tariff unduly lowered. A common reason 
for increased distance is the saving of the first cost of a very 
expensive although shorter line. 

(e) Finally, although there is a considerable and uncom- 
pensated loss resulting from curvature and grade which will 
justify a considerable expenditure to avoid them, there is by 
no means as much justification to incur additional expenditure 
to avoid distance. Of course needless lengthening should be 
avoided. A moderate expenditure to shorten the line may be 
justifiable, but large expenditures to decrease distance are 
never justifiable except when the great bulk of the traffic is 
exceedingly hea\^^ and is competitive. 

415. Justification of decreasing distance to save time. It 
should be recalled that the changes which an engineer may 
make which are physically or financially possible will ordi- 
narily have but little effect on the time required for a trip. 
The time which can thus be saved will have practically no value 
for the freight business — at least any value which would justify 
changing the route. When there is a large directly competitive 



§ 416. DISTANCE. 483 

passenger traffic between two cities {e.g. New York to Phila- 
delphia) a difference of even 10 minutes in the time required 
for a run might have considerable financial importance, but 
such cases are comparatively rare. It may therefore be con- 
cluded that the value of the time saved by shortening distance 
will not ordinarily be a justification for increased expense to 
accomplish it. 

416. Effect of change of distance on the business done. 
The above discussion is based on the assumption that the busi- 
ness done is unaffected b}^ any proposed change in distance. 
If a proposed reduction in distance involves a loss of business 
obtained, it is almost certainly unwise. But if by increasing 
the distance the original cost of the road is decreased (because 
the construction is of less expensive character) , and if the receipts 
are greater, and are increased still more by an increase in busi- 
ness done, then the change is probably wise. While it is almost 
impossible in a subject of such complexity to give a general 
rule, the following is generally safe: Adopt a route of such length 
that the annual traffic per mile of line is a maximum. This 
statement may be improved by allowing the element of original 
cost to enter and say, adopt a route of such length that the annual 
traffic per mile of fine divided by the average cost per mile is 
a maximum. Even in the above the operating cost per mile, 
as affected by the curvature and grades on the various routes, 
does not enter, but any attempt to formulate a general rule 
which would allow for variable operating expenses would e\'i- 
dently be too complicated for practical application. 



CHAPTER XXII. 
CURVATURE. 

417, General objections to curvature. In the popular mind 
curvature is one of the most objectionable features of railroad 
alignment. The cause of this is plain. The objectionable 
qualities are on the surface, and are apparent to the non-tech- 
nical mind. They may be itemized as follows: 

1. Curvature increases operating expenses by increasing (a) 
the required tractive force, (b) the wear and tear of roadbed 
and track, (c) the wear and tear of equipment, and (d) the 
required number of track- walkers and watchmen. 

2. It may affect the operation of trains (a) by limiting the 
length of trains, and (b) by preventing the use of the heaviest 
types of engines. 

3. It may affect travel (a) b}'- the difficulty of making time, 
(b) on account of rough riding, and (c) on account of the appre- 
hension of danger. 

4. There is actually an increased danger of collision, derail- 
ment, or other form of accident. 

Some of these objections are quite definite and their true 
value may be computed. Others are more general and vague 
and are usually exaggerated. These objections will be dis- 
cussed in inverse order. 

418. Financial value of the danger of accident due to curva- 
ture. At the outset it should be realized that in general the 
problem is not one of curvature vs. no curvature, but simply 
sharp curvature vs. easier curvature (the central angle remain- 
ing the same), or a greater or less percentage of elimination 
of the degrees of central angle. A straight road between ter- 
mini is in general a financial (if not a physical) impossibility. 
The practical question is then, how much is the financial value 
of such diminution of danger that may result from such elimi- 
nations of curvature as an engineer is able to make? 

484 



§ 419. CURVATURE. 485 

In the year 1898 there were 2228 railroad accidents reported 
by the Bailroad Gazette, whose Usts of all accidents worth re- 
porting are very complete. Of these a very large proportion 
clearly had no relation whatever to curvature. But suppose 
we assume that 50% (or 1114 accidents) were directly caused 
by curvature. Since there are approximately 200,000 curves 
on the railroads of the country, there was on the average an 
accident for every 179 curves during the year. Therefore we 
may say, according to the theory of probabilities, that the 
chances are even that an accident may happen on any particular 
curve in 179 years. This assumes all curves to be equally danger- 
ous, which is not true, but we may temporarily consider it to be 
true. If, at the time of the construction of the road, SI. 00 were 
placed at compound interest at 5% for 179 years, it would pro- 
duce in that time $620.89 for each dollar saved, wherewith to pay 
all damages, while the amount necessary to eliminate that cur- 
vature, even if it were possible, would probably be several thou- 
sand dollars. The number of passengers carried one mile for 
one killed in 1898-99 was 61,051,580. If a passenger were to 
ride continuously at the rate of sixty miles per hour, day and 
night, year after year, he would need to ride for more than 116 
years before he had covered such a mileage, and even then the 
probabilities of his death being due to curvature or to such a 
reduction of curvature as an engineer might accomplish are 
very small. Of course particular curves are often, for special 
reasons, a source of danger and justify the employment of 
special watchmen. They would also justify very large expen- 
ditures for their elimination if possible. But as a general 
proposition it is evidently impossible to assign a definite money 
value to the danger of a serious accident happening on a par- 
ticular curve which has no special elements of danger. 

Another element of safety on curved track is that trait of 
human nature to exercise greater care where the danger is more 
apparent. Many accidents are on record which have been 
caused by a carelessness of locomotive engineers on a straight 
track when the extra watchfulness usually observed on a curved 
track would have avoided them. 

419. Effect of curvature on travel, (a) Difficulty in making 
time. The growing use of transition curves has largely elimi- 
nated the necessity for reducing speed on curves, and even when 
the speed is reduced it is done so easily and quickly by means 



486 RAILROAD CONSTRUCTION. § 419. 

of air-brakes that but little time is lost. If two parallel lines 
were competing sharply for passenger traffic, the handicap of 
sharp curvature on one road and easy curvature on the other 
might have a considerable financial value, but ordinarily the 
mere reduction of time due to sharp curvature will not have any 
computable financial value. 

(b) On account of rough riding. Again, this is much reduced 
by the use of transition curves. Some roads suffer from a gen- 
eral reputation for crookedness, but in such cases the excessive 
curvature is practically unavoidable. This cause probably 
does have some effect in influencing competitive passenger 
traffic. 

(c) On account of the apprehension of danger. This doubtless 
has its influence in deterring travel. The amount of its influence 
is hardly computable. When the track is in good condition 
and transition curves are used so that the riding is smooth, 
even the apprehension of danger will largely disappear. 

Travel is doubtless more or less affected by curvature, but 
it is impossible to say how much. Nevertheless the engineer 
should not ordinarily give this item any financial weight what- 
soever. Freight traffic (two thirds of the total) is unaffected 
by it. It chiefly affects that limited class of sharply competi- 
tive passenger traffic — a traffic of which most roads have not a 
trace. 

420. Effect on operation of trains, (a) Limiting the length 
of trains. When curvature actually limits the length of trains, 
as is sometimes true, the objection is valid and serious. But 
this can generally be avoided. If a curve occurs on a ruling 
grade without a reduction of the grade sufficient to compensate 
for the curvature, then the resistance on that curve will be a 
maximum and that curve '^ill limit the trains to even a less 
weight than that which may be hauled on the ruling grade. 
In such cases the unquestionably correct policy is to ''com- 
pensate for curvature,'' as explained later (see §§ 427, 428), and 
not allow such an objection to exist. It is possible for curvature 
to limit the length of trains even without the effect of grade. 
On the Hudson River R. R. the total net fall from Albany to 
New York is so small that it has practically no influence in 
determining grade. On the other hand, a considerable portion 
of the route follows a steep rocky river bank which is so crooked 
that much curvature is unavoidable and very sharp curvature 



re J| 



§ 420. CURVATURE. 487 

can only be avoided by very large expenditure. As a consequence 
sharp curvature has been used and the resistance on the curves 
is far greater than that of any fluctuations of grade which it 
was necessary to use. Or, at least, a comparatively small 
expenditure would suffice to cut down any grade so that its 
resistance would be less than that of some curve which could 
not be avoided except at an enormous cost. And as a result, 
since the length of trains is really limited by curvature, minor 
grades of 0.3 to 0.5% have been freely introduced w^hich 
might be removed at comparatively small expense The above 
case is verN^ unusual. Low grades are usually associated with 
generally level country where curvature is easily avoided — 
as in the Camden and Atlantic R. R. Even in the extreme 
case of the Hudson River road the maximum curvature is 
only equivalent to a comparatively low ruling grade. 

(b) Preventing the use of the heaviest types of engines. The 
validity of this objection depends somewhat on the degree of 
curvature and the detailed construction of the engine. AMiile 
some types of engines might have difficulty on curves of ex- 
tremely short radius, yet the objection is ordinarily invalid. 
This will best be appreciated when it is recalled that the " Con- 
solidation" type was originally designed for use on the sharp 
curvature of the mountain divisions of the Lehigh Valley R. R., 
and that the type has been found so satisfactory that it has 
been extensively employed elsewhere. It should also be re- 
membered that during the Civil War an immense traffic daily 
passed over a hastily constructed trestle near Petersburg, Va., 
the track ha\dng a radius of 50 feet. As a result of a test made 
at Renovo on the Philadelphia and Erie R. R. by Mr. Isaac 
Dripps, Geu. Mast. Mech., in 1875,* it was claimed that a 
Consolidation engine encountered less resistance per ton than 
one of the ''American" type. Whether the test was strictly 
reliable or not, it certainly demonstrated that there was no 
trouble in using these heavy engines on very sharp curvature, 
and we may therefore consider that, except in the most extreme 
cases, this objection has no force whatsoever. 

* Seventh An. Rep. Am. Mast. Mech. Assn. 



488 



RAILROAD CONSTRUCTION. 



§ 421. 



EFFECT OF CURVATURE ON OPERATING EXPENSES. 

421. Relation of radius of curvature and of degrees of 
central angle to operating expenses. The smallest consideration 
will show that the sharper the curvature the greater will be 
the tractive force required, also the greater per unit of track 
length wdll be the rail wear and the general wear and tear on 
roadbed and rolling stock. But it would be inconvenient 
to use a relation between operating expenses and radius of 
curvature, because even when such a relation was found there 
would be two elements to consider in each problem — the radius 
and the length of the curve. The method w^hich will be here 
developed cannot claim to be strictly accurate or even strictly- 
logical, but, as will be shoA\Ti later, the most uncertain elements 
of the computation ha^'e but a small influence on the final 
result, and the method is in general the only possible method of 
solution. The outline of the method is as follows: 

(1) For reasons given in detail later, it is found that the 
expenses, wear, etc., on the track from A io B will be substan- 
tially the same whether by the route M or N. The wear, etc., 



Z"^^' 




A 



Fig. 208. 



per foot at .V is of course greater, but the length of curve is 
less. Therefore the effect of the curvature depends on the 
degrees of central angle ^ and is independent of the radius. 



§ 422. CURVATURE. 489 

(2) At what degree of curvature is the total train resistance 
double its value on a tangent? Probably no one figure would 
be exact for all conditions. Train resistance varies with the 
velocity and wdth the various conditions of train loading even 
on a tangent, and it is by no means certain (or even probable) 
that the ratio would be exactly the same for all conditions. 
As an average figure we may say that a train running at average 
velocity on a 10° curve will encounter a resistance due to cur- 
vature which is approximately equal to the average resistance 
found on a level tangent. On a 10° curve therefore the resist- 
ance is doubled. 

(3) A train-mile costs about so much — approximately $1.35 
Doubling the tractive resistance will increase certain items o\ 
expenditure about so much. Their combined value is so much 
j)er cent of the cost of a train-mile. A mile of continuous 10* 
curve contains 528° of central angle. A mile of such track 
would add so much per cent to the average train-mile expenses^ 
and each degree of central angle is responsible for ^|g- of this 
increase. Since the increase is irrespective of radius and de- 
pends only on the degrees of central angle, we therefore say 
that each degree of central angle of a curve vnll add so much 
to the average operating expenses of a train-mile. 

The "cost per train-mile" considered above should be con- 
sidered as the cost of a mile of level tangent. If we for a moment 
consider that all the railroads of the country were made abso- 
lutely straight and level, it is apparent that the average cost 
per train-mile instead of being about $1.35, would be somewhat 
less. The percentage should therefore be applied to this reduced 
value, but the net effect of this change would evidentl}^ be 
small. 

422. Effect of curvature on maintenance of way. A 
very large proportion of the items of expense in a train-mile 
are absolutely unaffected by curvature. It will therefore 
simphfy matters somewhat if we at once throw out all the un- 
affected items. Of the items of maintenance of way and struc- 
ture all but the first three will be thrown out. Item 4 will be 
somew^hat affected when bridges or trestles occur on a curve. 
But when it is considered what a very small percentage of this 
small item (2.460%) could be ascribed to curvature, since the 
very large majority of bridges and trestles are purposely made 



490 RAILROAD CONSTRUCTION. § 422. 

straight, and since culverts, etc., are not affected, we may 
evidently ignore any variation in the item. 

Item I. Repairs of Roadway. A very large proportion of the 
sub-items are absolutely unaffected. The care of embankments 
and slopes, ditching, weeding, etc., are evidently unaffected. 
The track labor on rails and ties and the work of surfacing 
will evidently be somewhat increased and yet it is very seldom 
that the length of a track section would be decreased simply 
on account of excessive curvature. But 528° per mile is an 
excessive amount of curvature. The average for the whole 
country is about 30° per mile, and there are very few instances 
of that amount of curvature (528°) in the length of a single mile. 
As before intimated, it is reasonable to assume that the extra 
work per foot on a 20° curve would be 10 times the extra work 
on a 2° curve, which verifies the general statement that the 
extra cost varies as the degrees of central angle. Considering 
how much of this item is independent of curvature and how 
little even the track labor is affected, it is possibly overstating 
the case to allow 25% increase for 528° of curvature in one 
mile. 

Item 2. Renewals of Rails. The excess wear due to cur- 
vature has never been determined with satisfactory conclu- 
siveness. Some tests have been made within the last few years 
on the Northern Pacific Railroad, which have perhaps followed 
the only practical method for determining the law of rail wear 
on curves. Selected rails on several tangents and curves of 
varying degrees of curvature were annually taken up, cleaned 
and weighed, and the annual loss due to wear was noted. The 
results indicated a loss of weight on curves varying nearly 
according to the degree of curve, and that the excess wear on 
a curve is 22.6% per degree of curve over that on a tangent. 
For a 10° curve, this would mean an excess wear of 226%. 

Item 3. Renewals of Ties. Curvature affects ties by in- 
creasing the ^'rail cutting'' and on account of the more frequent 
respiking, which ''spike-kills'' the ties even before they have 
decayed. Wellington estimates that a tie w^hich will last 
nine years on a tangent will last but six years on a 10° curv^e. 
He adds 50% for tie renewals. He considers the decrease in 
tie life to be proportional to the degree of curve and therefore 
again verifies the general statement made above regarding the 
expense of curvature. 



§ 424. CURVATURE. 491 

423. Effect of curvature on maintenance of equipment. 

Items 11, 16, 18, and 19 will be considered as unaffected. 

Item 12. Repairs and Renewals of Locomotives. Curves 
affect locomotive repairs by increasing very largely the wear 
on tires and wheels. We can also say that the additional power 
required on curves increases the strain and the wear and will 
thus increase the cost of repairs. It has been estimated that 
about i or 12.5% of the cost of engine repairs is due to the 
effect of curvature. On the basis that the average curvature 
of the roads of the country is about 35° per mile, then each 
degree of curvature would be responsible for 0.35% of the 
cost of engine repairs. Although it probably would not be true 
to say that a continuous 10° curve would increase the cost of 
engine repairs by 528 times this amount or by 185%, it is more 
reasonable to assume that the cost of engine repairs increases 
as the amount of curvature for the ordinary curvature as used. 
Since this value, as well as others, is to be divided by 528 to 
obtain the extra cost of one degree of curvature, we may add 
185% for this item. 

Items 13, 14, and 15. By a similar course of reasoning to 
that given above, the estimates for items 14 and 15 will be 
made 100%, while that for item 15 will be made only 50%, 
because such a large proportion of the expenses of item 13 are 
due to painting and maintaining upholstery, which have no 
relation to variations in alignment. 

Item 17. The repairs and renewals of shop machinery and 
tools will not be increased more than 50% per mile for the addi- 
tional repairs required of the above equipment. 

424. Effect of curvature on conducting transportation. 
We may at once throw out all items except 22, 23, 24, and 25, 
a small part of 28, and possibly 35; 36, and 37. This last group 
has already been discussed in §418; the aggregate of the three 
items is but 1.752%; curvature is responsible for only a small 
proportion of the item, and the reduction which an engineer 
is able to effect would be so small that we may neglect it. 

Item 28 is somewhat analogous to the above. Curvature does 
not affect a large part of the item, but an extreme case of curva- 
ture Tvdll occasionally require an extra watchman. Consider- 
ing, however, that curvature does not in general require watch- 



492 



RAILROAD CONSTRUCTION. 



§ 424. 



men, and that such cases are the unusual cases in mountainous 
regions where the curvature is unavoidable and not materially 
reducible, it would eiidently be wrong to charge curvature in 
general with such an item, although there would be justifica- 
tion for it in individual cases. It will therefore be ignored. 

Items 22, 23, 24, and 25. In § 407, Chapter XXI, the propor- 
tion of fuel assigned to direct hauling on a tangent is computed 
as amounting to about 55%. Since this direct resistance is as- 
sumed to be exactly doubled, we will charge 55% for fuel. There 
will e^ddently be no error worth considering in allowing the same 
proportionate amount as the charge for water, oil, waste, etc. 

^'General expenses," items 47 to 53, are of course unaffected. 

425. Estimate of total effect per degree of central anglco 
Compiling the above estimates we have the following tabulation ' 

TABLE XXII. — EFFECT ON OPERATING EXPENSES OF CHANGES 
IN CURVATURE. 



Item 


Normal 


Per cent 


Cost per mile, 


No 


average. 


affected. 


per cent. 


1 


10.782 


25 


2.70 


2 


1.403 


226 


3.17 


3 


2.868 


50 


1.43 


rtl 


5.943 








20.996 




7.30 




11 


.600 








12 


7.011 


185 


12.97 


13 


2.125 


50 


1.06 


14 


7.561 


100 


7.56 


15 


.222 


100 


.22 


16 


.216 








17 


.606 


50 


.30 


18 


.042 








19 


.542 








18.925 




22.11 




20 


1.772 








21 


9.527 








22 


10.571 


44 


4.65 


23 


.636 


44 


.28 


24 


.374 


44 


.16 


25 


.207 


44 


.09 


26 1 
46 / 


32.859 








55.946 




5.18 




47 \ 
53; 


4.133 










100 000 




34.59 







§ 425. CURVATURE, 493 

According to it, 528® of curvature in one mile would increase 
the expenses of each train passing over it by 34.59% of the 
average cost of a train-mile, and according to the general prin- 
ciples laid down in § 421, 1° of central angle of any curve, no 
matter what the radius, will increase the expenses by ^^-^ of 
34.59%, or .0655% per degree. Therefore the cost per year 
per daily train each way is (at 135c. per train-mile) 

135 X .0655% X 2 X 365 = 64.55c. 

As a simple illustration (a more extended one will be given 
later), suppose that by using greater freedom with regard to 
earthwork the crooked line sketched may be reduced to the 
simple curve shown and a curvature of, sa}^, 110° may be re- 
duced to, say, 60°. 

Note that since the extreme tangents are identical, the sav- 
ing in central angle results from the elimination of the reversed 




Fig. 209. 

curvature and of that part of the direct curv^ature necessary 
to balance the reversed curvature. Assume that there are six 
daily trains each way. Then the annual saving is 

50 X. 6455X6 = $193.65, 



494 RAILROAD CONSTRUCTION. § 425. 

which at 5% would justify an expenditure of $3873.00 If 
the extra cost of construction does not exceed this, the im- 
provement is justifiable, and is made all the more so if the proba- 
bilities are great that the future traffic will largely exceed six 
trains per day. At the same time the warning regarding ^^dis- 
counting the future" Tx-ith respect to expected traffic should 
not be neglected. The possible effect of change of distance 
has not been referred to in the above problem. In any case it 
is a distinct problem. According to the above sketch, the 
difference in distance is probably very slight, and consider- 
ing the compensating character of extra distance, such small 
differences may usually be disregarded. The possible effect 
of change of grade will be discussed in the next chapter. As- 
suming that there is no difference to be considered on account 
of either grade or distance, the question hinges on the advisa- 
bility of spending $3873.00 for the improvement. 

426. Reliability and value of the above estimate. It should 
be realized at the outset that no extreme accuracy is claimed 
for the above estimate. The effect of curvature is somewhat 
variable as well as uncertain, but such estimates have this great 
value. Vary the estimates of individual items as you please 
(within reason), and the final result is still about the same and 
may be used to guide the judgment. As an illustration, sup- 
pose that the item of renewals of rails is assumed to be affected 
300% rather than 226%, the justifiable expenditure to avoid 
the curvature in the above case may similarly be computed 
as $3989, an increase of about 3%. But, after all, the real 
question is not whether the improvement is worth $3873 or 
$3989. The extra work involved may perhaps be done for $500 
or it may require $10000. The above general method furnishes 
a criterion which, while not accurate, is so much better than a 
reliance on vague judgment that it should not be ignored. 



COMPENSATION FOR CURVATURE. 

427. Reasons for compensation. The effect of curvature on 
a grade is to increase the resistance by an amount which is equiv- 
alent to a material addition to that grade. On minor grades 
the addition is of little importance, but when the grade is nearly 
or quite the ruling grade of the road, then the additional resist- 
ance induced by a curve will make that curve a place of maxi- 



§427. 



CURVATURE. 



495 



mum resistance and the real maximum will be a 'S^irtual grade" 
somewhat higher than the nominal maximum. If, in Fig. 210, 




Fig. 210. 

AN represents an actual uniform grade consisting of tangents 
and curves, the ''virtual grade'' on curves at BC and DE may 
be represented by BC and DE, If BC and DE are very long, 
or if a stop becomes necessary on the curve, then the full dis- 
advantage of the curve becomes developed. If the whole grade 
may be operated without stoppage, then, as elaborated further 
in the next chapter, the whole grade may be operated as if equal 
to the average grade, AF, which is better than BC, although 
much worse than AN. The process of ''compensation" con- 
sists in reducing the grade on every curve by such an amount 
that the actual resistance on each curve, due to both curvature 
and grade, shall precisely equal the resistance on the tangent. 
The practical effect of such reduction is that the "virtual" grade 
is kept constant, while the nominal grade fluctuates. 

One effect of this is that (see Fig. 211) instead of accomplish- 




FiG. 211. 



ing the vertical rise from A to 6^ (i.e., HG) in the horizontal 
distance AH, it requires the horizontal distance AK. Such an 
addition to the horizontal distance can usually be obtained by 
proper development, and it should always be done on a ruling 



496 RAILROAD CONSTRUCTION. §427. 

grade. Of course it is possible that it will cost more to accom- 
plish this than it is worth, but the engineer should be sure of 
this before allowing this virtual increase of the grade. 

European engineers early realized the significance of unre- 
duced curvature and the folly of la^dng out a uniform ruling 
grade regardless of the curvature encountered. Curv^e compen- 
sation is now quite generally allowed for in this country, but 
thousands of miles have been laid out without any compensa- 
tion. A very common limitation of curvature and grade has 
been the alliterative figures G° curvature and 60 feet per mile 
of grade, either singl}' or in combination. Assuming that the 
resistance on a 6° curve is equivalent to a 0.3% grade (15.84 feet 
per mile), then a 6° curve occurring on a 60-foot grade would 
develop more resistance than a 7o-foot grade on a tangent. 
The '' mountain cut-off of the Lehigh Valley Railroad near 
Wilkesbarre is a fme example of a heavy grade compensated 
for curvature, and yet so iaicl out that the \4rtual grade is uni- 
form from bottom to top, r- distance of several miles. 

428. Tlie proper rate Ox conipensation. This evidently is the 
rate of rrade of vliicli 'dio resL:itance just equals the resistance 
due to the curve. Luw such L'CGistance is variable. It is greater 
as the velocity is aoV;cr[; i'ii &"; renerally about 2 lbs. per ton 
(equivalent to a 0.1 ^S S^f/JIc) per degree of curve when starting 
a train. On this account', uiio compensation for a curve which 
occurs at a knovii stopping-place for the heaviest trains should 
be 0.1% p<2^' cegrec of curve. The resistance is not even strictly 
proportional uO the degree of curvature, although it is usually 
considered uO be go. In fact most formulae for curve resistance 
are based 0:1 such r^ relation. But if the experimentally deter- 
mined resi.^tr.nces for lev/ curvatures are applied to the excessive 
curvature of the Ner/ York Elevated road, for example, the 
rules become ridiculous. On this account the compensation 
per degree of curve may be made less on a sharp curve than on 
an easy curve. The compensation actually required for very 
fast trains is less than for slow trains, say 0.02 or 0.03% per 
degree of curve; but since the comparatively slow and heavy 
freight trains are the trains which are chiefly limited by ruling 
grade, the compensation must be made with respect to those 
trains. From 0.04 to 0.05% per degree is the rate of compen- 
sation most usually employed for average conditions. Curves 
which occur below sl known stopping-place for all trains need 



§ 429. CURVATURE. 497 

not be compensated, for the extra resistance of the curve will 
be simply utilized in place of brakes to stop the train. If a curve 
occurs just above a stopping-place, it is very serious and should 
be amply compensated. Of course the down-grade traffic need 
not be considered. 

It sometimes happens that the ordinary rate of compensa- 
tion will consume so much of the vertical height (especially if 
the curvature is excessive) that a steeper through grade must 
be adopted than was first computed, and then the trains might 
stall on the tangents rather than on the curves. In such cases 
a slight reduction in the rate of compensation might be justi- 
fiable. The proper rate of compensation can therefore be 
estimated from the following rules : 

(1) On the upper side of a stopping-place for the heaviest 
trains compensate 0.10% per degree of curve. 

(2) On the lower side of such a stopping-place do not com- 
pensate at all. 

(3) Ordinarily compensate about 0.05% per degree of curve. 

(4) Reduce this rate to 0.04% or even 0.03% per degree 
of curve if the grade on tangents must be increased to reach 
the required summit. 

(5) Reduce the rate somewhat for curvature above 8° or 10°. 

(6) Curves on minor grades need not be compensated, unless 
the minor grade is so heavy that the added resistance of the 
curve would make the total resistance greater than that of the 
the ruling grade, or unless there is some ground to believe that 
the ruling grade may sometime be reduced below that of the 
minor grade under consideration. 

429. The limitations of maximum curvature. What is the 
maximum degree of curvature which should be allowed on any 
road? It has been shown that sharp curvature does not prevent 
the use of the heaviest types of engines, and although a sharp 
curve unquestionably increases operating expenses, the increase 
is but one of degree with hardly any definite limit. The general 
character of the country and the gross capital available (or 
the probable earnings) are generally the true criterions. 

A portion of the road from Denver to Leadville, Col., is an 
example of the necessity of considering sharp curvature. The 
traffic that might be expected on the line was so meagre and 
yet the general character of the country was so forbidding 
that a road built according to the usual standards would have 



498 RAILROAD CONSTRUCTION. § 429. 

cost very much more than the traffic could possibly pay for. 
The line as adopted cost about -120,000 per mile, and yet in a 
stretch of 11.2 miles there are about 127 curves. One is a 25° 
20' curve, twenty-four are 24° curves, twenty-five are 20° curves, 
and seventy- two are sharper than 10°. If 10° had been made 
the limit (a rather high limit according to usual ideas), it is 
probable that the line would have been found impracticable 
(except wdth prohibitive grades) unless four or five times as 
much per mile had been spent on it, and this would have ruined 
the project financially. 

For many years the main-line traffic of the Baltimore and 
Ohio R. R. has passed over a 300-foot curve (19° 10') and a 
400-foot curve (14° 22') at Harper's Ferry. A few years ago 
some reduction was made in this by means of a tunnel, but 
the fact that such a road thought it wise to construct and operate 
such curves (and such illustrations on the heaviest-traffic roads 
are quite common) shows how foolish it is for an engineer to 
sacrifice money or (which is much more common) sacrifice 
gradients in order to reduce the rate of curvature on a road 
w^high at its best is but a second- or third-class road. 

Of course such belittling of the effects of curvature may 
be (and sometimes is) carried to an extreme and cause an engi- 
neer to fail to give to curvature its due consideration. Degrees 
of central angle should always be reduced by all the ingenuity 
of the engineer, and should only be limited by the general rela- 
tion between the financial and topographical conditions of the 
problem. Easy curvature is in general better than sharp curva- 
ture and should be adopted when it may be done at a small 
financial sacrifice, especially since it reduces distance generally 
and may even cut down the initial cost of that section of the 
road. But large financial expenditures are rarely, if ever, jus- 
tifiable where the net result is a mere increase in radius without 
a reduction in central angle. An analysis of the changes which 
have been so extensively made during late years on the Penn. 
R. R. and the N. Y., N. H. & H. R. R. will show invariably a 
reduction of distance, or of central angle, or both, and perhaps 
incidentally an increase in radius of curvature. There are but 
few, if any, cases where the sole object to be attained by the 
improvement is a mere increase in radius. 



§ 429. CURVATURE. 499 

The requirements of standard M. C. B. car-couplers have 
virtually placed a limitation on the radius on account of the 
corners of adjacent cars striking each other on very sharp 
curves. This limitation has been crystallized into a rule on 
the P. R. R. that no curve, even that of a siding, can have a 
less radius than 175 feet, which is nearly the radius of a 33° . 
curve. Of course only the most peremptory requirements of 
yard work would justify the employment of such a radius. 



CHAPTER XXIII. 

GRADE. 

430. Two distinct effects of grade. The effects of grade on 
train expenses are of two distinct kinds ; one possible effect is ^ 
very costly and should be limited even at considerable expen- 
diture; the other is of comparatively little importance, its cost 
being slight. As long as the length of the train is not limited, 
the occurrence of a grade on a road simply means that the engine 
is required to develop so many foot-pounds of work in raising 
the train' so many feet of vertical height. For example, if a 
freight train weighing 600 tons (1,200,000 lbs.) climbs a hill 
50 feet high, the engine performs an additional work of creating 
60,000,000 foot-pounds of potential energ}^ If this height is 
surmounted in 2 miles and in 6 minutes of actual time (20 
miles per hour), the extra work is 10,000,000 foot-pounds per 
minute, or about 303 horse-power. But the disadvantages of 
such a rise are always largely compensated. Except for the fact 
that one terminus of a road is generally higher than the other, 
every up grade is followed, more or less directly, by a do^Yn grade 
which is operated partly by the potential energy acquired during 
the previous climb. But w^hen we consider the trains running 
in both directions even the difference of elevation of the termini 
is largely neutralized. If we could eliminate frictional resist- 
ances and particularly the use of brakes, the net effect of minor 
grades on the operation of minor grades in both directions would 
be zero. Whatever was lost on any up grade would be regained 
on a succeeding do^Ti grade, or at any rate on the return trip. 
On the very lowest grades (the limits of which are defined later) 
we may consider this to be literally true, Adz., that nothing is 
lost by their presence; w^hatever is temporarily lost in climbing 
them is either immediately regained on a subsequent light down 
grade or is regained on the return trip. If a stop is required 
at the bottom of a sag, there is a net and uncompensated loss 
of energy. 

500 



§ 431. GRADE. 501 

On the other hand, if the length of trains is Umited by the 
grade, it will require more trains to handle a given traffic. The 
receipts from the traffic are a definite sum. The cost of hand- 
ling it will be nearly in proportion to the number of trains. 
Anticipating a more complete discussion, it may be said as an 
example that increasing the ruling grade from 1.20% (63.36 
feet per mile) to 1.55% (81.84 feet per mile — an increase of 
about 18.5 feet per mile) will be sufficient to increase the re- 
quired number of trains for a given gross traffic about 25%, 
i.e., five trains will be required to handle the traffic which four 
trains would have handled before at a cost slightly more than 
four-fifths as much. The effect of this on dividends may readily 
be imagined. 

431. Application to the movement of trains of the laws of 
accelerated motion. When a train starts from rest and acquires 
its normal v^'locity, it overcomes not only the usual tangent 
resistances (and perhaps curve and grade resistances), but it 
also performs work in storing into the train a vast fund of kinetic 
energy. This work is not lost, for every foot-pound of such 
energy may later be utilized in overcoming resistances, pro- 
vided it is not wasted by the action of train-brakes. If for a 
moment we consider that a train runs without any friction, 
then, when running at a velocity of v feet per second, it possesses 
a kinetic energy which would raise it to a height h feet, when 

h = — , in which g is the acceleration of gravity = 32.16. Assum- 
ing that the engine is exerting just enough energy to overcome 
the frictional resistances, the train would climb a grade until the 
train was raised h feet above the point where its velocity was v. 
When it had climbed a height h^ (less than h) it would have a 
velocity v^=\/2g{h — h'). As a numerical illustration, assume 

-y =30 miles per hour = 44 feet per second. Then /?- = — =30.1 feet, 

and assuming that the engine was exerting just enough force 
to overcome the rolling resistances on a level, the kinetic 
energy in the train would carry it for two miles up a grade of 
15 feet per mile, or half a mile up a grade of 60 feet per mile. 
When the train had climbed 20 feet, there would still be 10.1 
feet left and its velocity would be i'i=As/2gr (10.1) =25.49 feet 
per second = 17.4 miles per hour. These figures, however, must 
be slightly modified on account of the weight and the revolving 



532 RAILROAD CONSTRUCTION. § 432. 

action of the wheels, which form a considerable percentage 
of the total weight of the train. When train velocit}- is being 
acquired, part of the work done is spent in imparting the energy 
of rotation to the driving-wheels and various truck-wheels of 
the train. Since these wheels run on the rails and must turn 
as the train moves, their rotative kinetic energy is just as effec- 
tive — as far as it goes — in becoming transformed back into 
useful work. The proportion of this energy to the total kinetic 
energy has already been demonstrated (see Chapter XVI, 
§ 347). The value of this correction is variable, but an average 
value of 5% has been adopted for use in the accompanying 
tabular form (Table XXIII), in which is given the corrected 
''velocit}^ head'^ corresponding to various velocities in miles 
per hour. The table is computed from the following formula : 

^^ , .^ , , r^ in ft. per sec. 2.151vMnm.perh. ^^oo^^ 2 
Velocity head = gj^^ = ^^32 =0.03344?;^ 

adding 5% for the rotative kinetic energy of the wheels, 0.00167^;^ 



The corrected velocity head therefore equals 0.0351 Ij;^ 

Part of the figures of Table XXIII were obtained by inter- 
polation and the final hundredth may be in error b}^ one unit, 
but it may readily be showTi that the final hundredth is of no 
practical importance. It is also true that the chief use made 
of this table is with velocities much less than 50 miles per hour. 
Corresponding figures may be obtained for higher velocities, if 
desired, by multiplying the figure for half the velocity by jour. 

432. Construction of a virtual profile. The following simple 
demonstration will be made on the basis that the ordinary 
tractive resistances and also the tractive force of the locomo- 
tive are independent of velocity. For a considerable range of 
velocity which includes the most common freight-train velocities 
this assumption is so nearly correct that the method will give 
an approximately correct result, but for higher velocities and 
for more accurate results a more complicated method (given 
later) must be used. The following demonstration will serve 
well as a preliminary to the more accurate method. It may 
best be illustrated by considering a simple numerical example. 

Assuming that a train is passing A (see Fig. 212), running at 
30 miles per hour. Assume that the throttle is not changed or 
any brakes applied, but that the engine continues to exert the 



§432. 



GRADE. 



503 



TABLE XXUI. — VELOCITY HEAD (REPRESENTING THE KINETIC 
energy) of TRAINS MOVING AT VARIOUS VELOCITIES. 



Vel. 






















mi. 
hr. 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


10 


3.51 


3.58 


3.65 


3.72 


3.79 


3.87 


3.95 


4.02 


4.10 


4.17 


11 


4.25 


4.33 


4.41 


4.49 


4.57 


4.65 


4.73 


4.81 


4.89 


4.97 


12 


5.06 


5.15 


5.23 


5.32 


5.41 


5.50 


5.58 


5.67 


5.75 


5.84 


13 


5.93 


6.02 


6.12 


6.21 


6.31 


6.40 


6.50 


6.59 


6.69 


6.78 


14 


6.88 


6.98 


7.08 


7.19 


7.29 


7.39 


7.49 


7.60 


7.70 


7.80 


15 


7.90 


8.00 


8.11 


8.22 


8.33 


8.44 


8.55 


8.66 


8.77 


8.88 


16 


8.99 


9.10 


9.21 


9.32 


9.43 


9.55 


9.67 


9.79 


9.91 


10.03 


17 


10.15 


10.27 


10.39 


10.51 


10.63 


10.75 


10.87 


10.99 


11.12 


11.25 


18 


11.38 


11.50 


11.63 


11.76 


11.89 


12.02 


12.15 


12.28 


12.41 


12.55 


19 


12.68 


12.81 


12.95 


13.08 


13.22 


13.35 


13.49 


13.63 


13.77 


13.91 


20 


14.05 


14.19 


14.33 


14.47 


14.61 


14.75 


14.89 


15.04 


15.19 


15.34 


21 


15.49 


15.64 


15.79 


15.94 


16.09 


16.24 


16.39 


16.54 


16.69 


16.84 


22 


17.00 


17.15 


17.30 


17.46 


17.62 


17.78 


17.94 


18.10 


18.26 


18.42 


23 


18.58 


18.74 


18.90 


19.06 


19.22 


19.38 


19.55 


19.72 


19.89 


20.06 


24 


20.23 


20.40 


20.57 


20.74 


20.91 


21.08 


21.25 


21.42 


21.59 


21.77 


25 


21.95 


22.12 


22.30 


22.48 


22.66 


22.84 


23.02 


23.20 


23.38 


23.56 


26 


23.74 


23.92 


24.10 


24.28 


24.46 


24.65 


24.84 


25.03 


25 22 


25.41 


27 


25.60 


25.79 


25.98 


26.17 


26.36 


26.55 


26.74 


26.93 


27.13 


27.33 


28 


27.53 


27.73 


27.93 


28.13 


28.33 


28.53 


28.73 


28.93 


29.13 


29.33 


29 129.53 


29.73 


29.93 


30.13 


30.34 


30.55 


30.76 


30.97 


31.18 


31.39 


30 31.60 


31.81 


32.02 


32.23 


32.44 


32.65 


32.86 


33.08 


33.30 


33.52 


31 


33.74 


33.96 


34.18 


34.40 


34.62 


34.84 


35.06 


35.28 


35.50 


35.72 


32 


35.95 


36.17 


36.39 


36.62 


36.85 


37.08 


37.31 


37.54 


37.77 


38.00 


33 


38.23 


38.46 


38.69 


38.92 


39.15 


39.38 


39.62 


39.86 


40.10 


40.34 


34 


40.58 


40.82 


41.06 


41.30 


41.54 


41.78 


42.02 


42.26 


42.51 


42.76 


35 


43.01 


43.26 


43.51 


43.76 


44.01 


44.26 


44.51 


44.76 


45.01 


45.26 


36 


45.51 


45.76 


46.01 


46.26 


46.52 


46.78 


47.04 


47.30 


47.56 


47.82 


37 


48.08 


48.34 


48.60 


48.86 


49.12 


49.38 


49.64 


49.91 


50.18 


50.45 


38 


50.72 


50.99 


51.26 


51.53 


51.80 


52.07 


52.34 


52.61 


52.88 


53.15 


39 


53.42 


53.69 


53.96 


54.23 


54.51 


54.79 


55.07 


55.35 


55.63 


55.91 


40 


56.19 


56.47 


56.75 


57.03 


57.31 


57.59 


57.87 


58.16 


58.45 


58.74 


41 


59.03 


59.32 


59.61 


59.90 


60.19 


60.48 


60.77 


61.06 


61.35 


61.64 


42 


61.94 


62.23 


62.52 


62.82 


63.12 


63.42 


63.72 


64.02 


64.32 


64.62 


43 


64.92 


65.22 


65.52 


65.82 


66.12 


66.43 


66.74 


67.05 


67.36 


67.67 


44 


67.98 


68.29 


68.60 


68.91 


69.22 


69.53 


69.84 


70.15 


70.46 


70.78 


45 


71.10 


71.42 


71.74 


72.06 


72.38 


72.70 


73.02 73.34 


73.66 


73.98 


46 


74.30 


74.62 


74.94 


75.26 


75.59 


75.92 


76.25 76.58 


76.91 


77.24 


47 


77.57 


77.90 


78.23 


78.56 


78.89 


79.22 


79.55 79.89 


80.23 


80.57 


48 


80.91 


81.25 


81.59 


81.93 


82.27 


82.61 


82.95 83 29 


83.63 


83.97 


49 


84.32 


84.66 


85.00 


85.34 


85.69 


86.04 


86.39 86.74 


87.09 


87.44 


50 


87.79 


88.14 


88.49 


88.85 


89.20 


89.55 


89.91 


90.26 1 


90.61 


90.97 



same draw-bar pull. At A its ^Welocity head'' is that due to 30 
miles per hour, or 31.60 feet. At B it has gained 40 feet more, 
and its velocity is that due to a velocity head of 71.60 feet, or 
slightly over 45 miles per hour. At J5' its velocity is again 30 
miles per hour and velocity head 31.60 feet. At C the velocity 
head is but 6.60 feet and the velocity about 13.7 miles per hour. 



504 



RAILROAD CONSTRUCTION. 



§432. 



As the train runs from C to D its velocity increases to 30 miles at 
C and to over 45 miles per hour at D. At E the velocity again 
becomes 30 miles per hour. Although there will be some slight 
modifications of the above figures in actual practice, yet the above 
is not a fanciful theoretical sketch. Thousands of just such 
undulations of grade are daily operated in such a way, without 
disturbing the throttle or applying brakes, and the draw-bar 
.pull, if measured by a dynamometer, would be found to be 
practically constant. Of course the above case assumes that 

VTrtuaf profTle 




D'„,_^-^ 



Fig. 212. 
there are no stoppages and that the speed through the sags is 
not so great that safety requires the application of brakes. 
Observe that the ''virtual profile" is here a straight line — as it 
always is when the draw-bar pull is constant. The virtual 
profile (in this case as well as in every other case, illustrations 
of which will follow) is found by adding to the actual profile 
at any point an ordinate which represents the ''velocity head'' 
due to the velocity of the train at that point. 

As another case, assume that a train is climbing the grade AE 
and exerting a pull just sufficient to maintain a constant velocity 

up that grade. Then A'B' (parallel 
to AB) is the virtual profile, AA^ 
representing the velocity head. A 
stop being required at C, steam is 
shut off and brakes are applied 
at B, and the velocity head BB^ 
reduces to zero at C. The train 
starts from C, and at D attains a velocity corresponding to 
the ordinate DD' . At D the throttle may be slightly closed 
so that the velocity will be uniform and the virtual grade is 
D'E', parallel to DE. 

From the above it may be seen that a virtual profile has the 
following properties: 

(a) When the velocity is uniform, the virtual profile is parallel 
with the actual. 




Fig. 213. 



§ 433. GRADE. 505 

(b) When the velocity is increasing the profiles are separating; 
when decreasing the profiles are approaching. 

(c) When the velocity is zero the profiles coincide. 

433. Use, value, and possible misuse. The essential feature 
respecting grades is the demand on the locomotive. From the 
foregoing it may readily be seen that the ruling grade of a road 
is not necessarily the steepest nominal grade. W^hen a grade 
may be operated by momentum, i.e., when every train has an 
opportunity to take ^'sl run at the hill," it may become a very 
harmless grade and not limit the length of trains, while another 
grade, actually much less, which occurs at a stopping-place 
for the heaviest trains, will require such extra exertion to get 
trains started that it may be the worst place on the road. There- 
fore the true way to consider the value of the grade at any criti- 
cal place on the road is to construct a virtual profile for that 
section of the road. The required length of such a profile is 
variable, but in general may be said to be limited by points on 
each side of the critical section at which the velocity is definite, 
as at a stopping-place (velocity zero), or a long heavy grade where 
it is the minimum permissible, say 10 or 15 miles per hour. 

Since the velocities of different trains vary, each train will 
have its own virtual profile at any particular place. The fast 
passenger trains are generally unaffected, practically. The 
requirement of high average speed necessitates the use of power- 
ful engines, and grades which would stall a heavy freight will 
only cause a momentary and harmless reduction of speed of 
the fast passenger train. 

A possible misuse of virtual profiles lies in the chance that a 
station or railroad grade crossing may be subsequently located 
on a heavy grade that w^as designed to be operated by momen- 
tum. But this should not be used as an argument against the 
employment of a virtual profile. The virtual profile shows the 
actual state of the case and only points out the necessity, if an 
unexpected requirement for a full stoppage of trains at a critical 
point has developed, of changing the location (if a station), or 
of changing the grade by regrading or by using an overhead 
crossing. Examples of such modifications are given in Chap- 
ter XXIV, The Improvement of Old Lines. 

434. Undulatory grades. Advantages. Money can generally 
be saved by adopting an actual profile which is not strictly 
uniform — the matter of compensation for curvature being here 



506 RAILROAD CONSTRUCTION. § 434. 

ignored. Its effect on the operation of trains is harmless pro- 
vided the sag or hump is not too great. In Fig. 214 the undu- 
latory grade may actuall}^ be operated as a uniform grade AG. 
The sag at C must be considered as a sag, even though BC is actu- 
ally an up grade. But the engine is supposed to be working 




Fig. 214. 
hard enough to carry a train at uniform velocity up a grade AG, 
Therefore it gains in velocity from B to C, and from C toD loses 
an equal amount. It may even be proven that the time re- 
quired to pass the sag will be slightly less than the time required 
to run the uniform grade. 

Disadvantages. The hump at F is dangerous in that, if the 
velocity at E is not equal to that corresponding to the extra 
velocity-head ordinate at -F, the train will be stalled before 
reaching F. In practice there should be considerable margin. 
Any train should have a velocity of at least 10 miles per hour 
in passing any summit. This corresponds to a velocity head 
of 3.51 feet. An extra heavy head \vind, slippery rails, etc., 
may use up any smaller margin and stall the train. If the 
grade AG is a ruling grade, then no hump should be allowed 
under any circumstances. For the heaviest trains are supposed 
to be so made up that the engine will just haul them up the 
ruling grades — of course with some margin for safety. Any 
mcrease of this grade, however short, would probably stall the 
train. 

Safe limits. It is quite possible to have a sag so deep that 
it is not safe to allow freight trains to rush through them with- 
out the use of brakes. The use of brakes of course adds a 
distinct element of cost. To illustrate: If a freight train is 
running at a velocity of 20 miles per hour (velocity head 14.05 
feet) and encounters a sag of 25 feet, the velocity head at the 
bottom of the sag will be 39.05 feet, which corresponds to a 
velocity of 33.3 miles per hour. This approaches the limit of 
safe speed for freight trains, and certainly passes the limit for 
trains not equipped with air-brakes and automatic couplers. 



§ 435. GRADE. 507 

The term ''safe limits '' as used here, refers to the limits within 
which a freight train may be safely operated without the appli- 
cation of brakes or varying the work of the engine. Of course 
much greater undulations are frequently necessary and are 
safely operated, but it should be remembered that they add a 
distinct element to the cost of operating trains and that they 
must not be considered as harmless or that they should be 
introduced unless really necessary. 



MINOR GRADES. 

435. Basis of cost of minor grades. The basis of the com- 
putation of this least objectionable form of grade is as follows: 
The resistance encountered by a train on a level straight track 
is somewhat variable, depending on the velocity and the num- 
ber and character of the cars, but for average velocities we 
may consider that 10 lbs. per ton is a reasonable figure. This 
value agrees fairly well with the results of some dynamometer 
tests made by Mr. P. H. Dudley, using a passenger train of 
313 tons running at about 50 miles per hour. It also agrees 
with the Engineering News formula (Eq. 143) for the re- 
sistance of a train at a velocity of 32 miles per hour. Ten 
pounds per ton is the grade resistance of a 0.5% grade, or 
that of 26.4 feet per mile. On the above basis, a 0.5% grade 
mil just double the tractive resistance on a level straight track. 
We may compute, as in the previous chapter, the cost of doubling 
the tractive resistance for one mile. But since the extra resist- 
ance is due to lifting the train through 26.4 feet of elevation, 
w^e may divide the extra cost of a mile of 0.5% grade by 26.4 
and we have the cost of one foot of difference of elevation, and 
then (disregarding the limiting effect of grades) we may say that 
this cost of one foot of difference of elevation will be independent 
of the rate of grade. There are, however, limitations to this 
general proposition which will be developed in the next section. 

436. Classification of minor grades. These are classified with 
reference to their effect on the operation of trains. In the first 
class are grades which may be operated without changing the 
work of the engine and which have practically no other effect 
than a harmless fluctuation of the velocity. But a grade which 
belongs to this class when considering a fast passenger train Tvdll 
belong to another class when considering a slow but heavy 




508 RAILROAD CONSTRUCTION. § 436. 

freight train. And since it is the slow heavy freight trains 
which must be chiefl}^ considered, a grade will usually be classi- 
fied with respect to them. The limit of class A (the harmless 

class) therefore depends on the 
n' ^^^y^^ maximum allowable speed. The 

effect of a sag on speed will depend 
'^^---^^^Jy^ ^^ the A'ertical feet of drop rather 

than on the rate of grade, for 
„ with the engine working as usual 

on even a light down grade a train 
would soon exceed permissible speed. Assume that a freight 
train runs at an average speed of 15 miles per hour with a 
minimum of 10 miles and a permissible maximum of 30 miles 
per hour. Assume that a train runs up the grade at A with 
a uniform velocity of 15 miles per hour, i.e., the engine is 
working so that the velocity would be uniform to C. How 
much sag (BB') can there be without the speed exceeding 30 
miles per hour? 

Velocity head for 30 miles per hour 31 . 60 

" '' 15 " '' '' 7.90 

The drop BB' will therefore be 23 . 70 

While each case must be figured by itself, considering the 
probable velocity of approach and the maximum permissible 
velocity, we may say that a sag of about 24 feet will ordinarily 
mark the limit of this class. With a higher velocity of approach 
even this limit will be much reduced. 

The classification therefore applies to sags and humps and 
to the vertical feet of drop or climb which are involved, rather 
than to grade per se. The practical application of these prin- 
ciples is necessarily confined to humps or sags which are pos- 
sibly removable and does not apply to the long grades which 
are essential to connect predetermined points of the route — 
grades which are irreducible except by development and which 
must be studied as ruling grades (see §§ 440-445). 

The application therefore consists in the comparative study 
of two proposed grades, noting the relative energy required 
to operate them and the probable cost. The depth in feet 
saved would be the maximum difference between the grades, 
and the classification will depend on the necessary method of 
operating the trains. 



§ 437. GRADE. 509 

The next classification (B) applies to drops so deep that 
steam must be shut off when descending the grade, while the 
work required of the engine when ascending the opposite grade 
is correspondingly increased. The loss is not so serious as 
in the next case, but the inability of the engine to w^ork con- 
tinuously may result in a failure to accumulate sufficient kinetic 
energy to carry the train over a succeeding summit. 

The third class (C) includes the grades so long that brakes 
must be applied to prevent excessive velocity. The loss in- 
volved is very heavy; the brakes require power for their appli- 
cation, they wear the brake-shoes and wheel-tires, they destroy 
kinetic or potential energy which had previously been created, 
w^hile the tax on the locomotive on the corresponding ascend- 
ing grade is very great. The ascending grade may or may not 
be a ruling grade. 

437. Effect on operating expenses. As in Chapter XXII 
we may at once throw out a large proportion of the items of 
expense of an average train-mile. In " maintenance of w^ay 
and structures" items 4 to 10 are evidently unaffected. 

Item I. Repairs of Roadway. It is very plain that a 
large proportion of the sub-items are absolutely unaffected 
])y minor grades. In fact it is a little difficult to ascribe any 
definite increase to any sub-item. The rail wear is somew^hat 
increased and this will have some effect on the trackwork, 
but on the other hand the increased grade sometimes results 
in better drainage and therefore less work to keep the track 
in condition. Wellington allows 5% increase as a ^'liberal 
estimate" for class C, and no increase for the other classes. 

Item 2. Renewals of Rails. Observations of rail w^ear 
on heavy grades show that it is much greater than on level 
tangents. But usually such heavy grades are operated by 
shorter trains or with the help of pusher engines, and the pro- 
portion of engine tonnage to the total is much greater than is 
ordinarily the case And since an engine has much greater 
effect on rail wear than cars, particularly on account of the 
use of sand, an excess of engine tonnage would have a marked 
effect. But such circumstances would inevitably accompany 
ruling grades and not minor grades. Nevertheless the effect 
of the use of sand on up grades and the possible skidding of 
wheels on down grades will wear the rails somew^hat. Even 
the possible slipping of the drivers, although sand is not used, 



510 RAILROAD CONSTRUCTION. § 438. 

will wear the rails. Wellington allows 10% increase for class 
C and 5% for class B. 

Item 3. Renewals of Ties. The added wear of ties might 
be considered proportional to that of the rails except that, as 
in the case of the roadbed in general, the better drainage secured 
by the grade will tend to increase the life of the ties. Welling- 
ton makes the estimate the same as for item 1, 5% for class C 
and no increase for the other classes. 

Maintenance of equipment. Items 11, 16, 17, 18, and 10 
are evidently unaffected. Items 12 to 15. The chief sub- 
items of increase will evidently be the repairs and renewals of 
wheels and brake-shoes both for locomotives and cars. In 
the case of cars the draw-bar is apt to suffer from severe alter- 
nate compression and extension due to push and pull. The 
locomotive mechanism will suffer somewhat from the extra 
demands on it, and the boiler on account of the intermittent 
character of the demands on it. It would seem as if such 
effects would be quite large, but an examination of the com- 
parative records of engine and car repairs on mountain divisions 
and on comparatively level divisions shows no such difference 
as might be expected. On this account Wellington cuts down 
these items to 4% for class C and 1% for class B. 

Conducting transportation. As in Chapter XXI, § 407, since 
the resistance is assumed to be doubled, we may take the same 
figure (44%) as the cost of the fuel for climbing the 26.4 feet. 
But the total cost of both the rise and fall is to be considered. 
In class B, although steam is shut off, heat (and fuel) is wasted 
by mere radiation. This has been estimated (Chapter XXI, 
§ 407) as about 5%. Therefore we may allow 49% for class B, 
For class C we must allow in addition the energy spent in 
applying brakes, which we may assume as 5% more, making 
54%. Items 23, 24, and 25 may be estimated similarly. The 
other items under this head as well as General Expenses are 
evidently unaffected. 

438. Estimate of the cost of one foot of change of elevation. 
Collecting these estimates, we have the accompanying tabular 
form, showing that the percentage of increase for operating 
grades of class B or class C will be 6.08% and 7.92%, respect- 
ively. On the basis of an average cost of 135c. per train-mile, 
the additional cost for the 26.4 feet in «ne mile would be 8.21 c. 
and 10.70 c, or 0.311 c. and 0.405 c. per foot. For each train per 



§ 438. 



GRADE. 



611 



day each way per year the value per foot of difference of ele- 
vation is: 

For class B: 2 X 365 X $0.00311 = $2.27; 
'' '' C: 2 X 365 X $0.00405 = $2.96. 

TABLE XXIV. EFFECT ON OPERATING EXPENSES OF 26.4 FEET 

OF RISE AND FALL. 





Item (abbreviated).* 


Normal 
average. 


Class B. 


Class C. 


No. 


Per cent 
affected 


Cost 
per mile 


Per cent 
affected 


Cost 
per mile 


1 


Roadway 


10.782 
1.403 
2.868 
5.943 



5 







.07 






5 

10 
5 




.54 


2 


Rails 


14 


3 


Ties 


.14 


4-10 


Bridges, buildings, etc. 
Maintenance of way . 







20.996 




.07 




.82 


11 
12 
13 
14 
15 
16 
17 


Superintendence 

Repairs locomotives . . . 

Repairs pass, cars 

Repairs freight cars . . . 
Repairs work cars .... 
^larine equipment .... 
Shops 


.600 

7.011 

2 . 125 

7.561 

.222 

.216 

.606 

.042 

.542 




1 
1 
1 
1 










.07 

.02 

.08 

.00 











4 
4 
4 
4 







.28 
.08 
.30 
.01 




18 
19 


Stat, and printing .... 
Other expenses ....... 

Main, of equip 








18.925 




.17 





.67 


20 
21 


Superintendence 

Enginemen 


1.772 

9.527 

10.571 

.636 

.374 

.207 

32.859 

55 . 946 




49 
49 
49 
49 








5.25 

.31 

.18 
.10 






54 
54 
54 
54 








02 


Fuel 


5.78 


23 


Water 


34 


• 24 


Oil, etc 


.20 


. 25 
26-46 


Other supplies 

Train service, station 
seivice, etc 


.11 





Conducting transp. . . 








5.84 




6.43 


47-53 


General expenses 


4.133 


















100.000 




6.08 




7.92 



* For full title of item see Table XX. 



It will frequently happen that a grade must be considered as 
belonging to class C for heavy freight trains, and that it belongs 
to class B or even class A for other trains. If no Sunday trains 
are run, 313 should be used instead of 365 as a multiplier in 
the above equations. 



512 



RAILROAD CONSTRUCTION. 



§439. 



439. Operating value of the filling of a sag in a grade- Assume 
l^hat the sag is 4000 feet long and that its depth in the center 
is 35 feet, as sketched in Fig. 216. Assume that a freight train 
is approaching the sag from the right-hand side (running to the 
left), the speed of passing D being 25 miles per hour. The 0.3% 
grade to the left will furnish a gravity pull of 6 pounds per ton, 
which may be more than half the force required to pull the train, 
and the locomotive would have but little to do, even if there 
were no sag, to maintain the speed of 25 miles per hour. Assum- 
ing that the locomotive is doing just this amount of work in 
running to the left through the sag, it will gain in velocity. 
Its velocity head at D is that corresponding to 25 miles per 
hour, or 21.95 feet. Adding 35 feet, the depth of the sag, 
we have 56.95 feet, which is the velocity head at C, which 




means that the velocity of the train would be 40.3 miles per 
hour. For passenger trains this will not be an objectionably 
high velocity, and even freight trains, which are provided with 
air-brakes and standard M. C. B. couplers, which are now 
nearly universal, may be safely run at this velocity. Therefore 
for all trains which may run at a speed of over 40 miles per 
hour, this sag will belong to class A, or the harmless class, so 
far as trains moving to the left are concerned. The effect on 
trains moving to the right will depend partly on the rate of 
ruling grade on that section of the road. The conditions of 
class A pre-suppose that the draw-bar pull is constant, but horse- 
power is measured by the product of pull times velocity. As- 
sume that A 5 is a ruling grade — 0.3% has recently been adopted 
as ruling grade for revision work on the Erie R. R. Then, if 
20 miles per hour were the speed of approach (running to the 
right) the speed at C would have to be 37.4 miles per hour. 
But since the draw-bar pull is to be constant, the horse-power 



439. GRADE. 513 



m- 



developed would have to be nearly doubled ( — — j . Since 

this would be impossible, it would mean that such a sag would 
not only be a serious matter but would be prohibitive on a 
ruling grade. For ordinary roads, 0.3% will not be a ruling 
grade, and the possibility of temporarily increasing the horse- 
power developed by the locomotive while running through the 
sag, so that the draw-bar pull will remain constant, will be far 
greater. The ability to develop such horse-power is very apt 
to be the criterion as to whether a sag belongs to class A rather 
than the danger that the speed may be prohibitive. The crite- 
rion as to whether the grade belongs to class B or C for trains 
moving to the left, depends on whether brakes must be applied 
before reaching the bottom of the sag. A sharp curve at or 
near C might require the use of brakes to prevent a dangerous 
velocity. For trains moving to the right, there is no definite 
criterion between classes B and C, but a 2.05% grade is a very 
severe tax on a locomotive, especially when the assistance of 
momentum has been wasted by shutting off steam or by the 
applicatioii of brakes. Ignoring the possible limiting effect of 
a ruling grade (which is a separate matter) the value of the 35- 
foot sag is evidently 

35X$2.27 = S79.45 per daily round-trip train for class B 
and 

35 X $2.96 = $103.60 per daily round-trip train for class C, 

Assume that there are six daily trains each way, for which 
the grade would be classified as grade B, and four others, for 
which the sag would be classified as involving class C grade. 
Then on the above basis, the total annual cost would be 

6X $79. 45 = $476. 70 
4 X $103. 60 = $414. 40 

$891.10 

This annual cost capitalized at 5% equals $17,822, which is 
the justifiable expenditure to fill up the sag. The amount of 
fill in such a sag, roughly calculated, is 125,000 cubic yards. 
Assuming that it would cost 30 c. per cubic yard to make the 



514 RAILROAD CONSTRUCTION. § 439. 

fill, this would require an expenditure of $37,500, which appar- 
ently would not be justifiable. 

But another solution may be considered. It has been shown 
above that a sag may be made harmless for all classes of trains 
provided the depth is not greater than some uncertain limit, 
which depends on the particular circumstances of the case. 
The volume of earth in this fill is great on account of the great 
ilepth in the center. It may be readily computed that by fill- 
ing up the lower part of the sag so that its maximum depth 
below the grade line, BED, is about one-half of CE, or about 
17 feet, that the amount of earthwork required will be only 
about 25,000 yards rather than 125,000 yards. This would 
make the cost of such a fill practically $7500 rather than $37,500. 
If it could be shown that a sag of about 18 feet could be operated 
on the class A basis by all trains, then it would certainly be 
justifiable to expend S7500 in order to secure a reduction in the 
operating expenses, whose capitalized value according to the 
above calculation, is $17,822. 



RULING GRADES. 

440. Definition. Ruling grades are those which limit the 
weight of the train of cars which may be hauled by one engine. 
The subject of '^ pusher grades" will be considered later. For 
the present it will suffice to say that on all well-designed roads 
the large majority of the grades on any one division are kept 
below some limit w^hich is considered the ruling grade. If a 
heavier grade is absolutely necessary no special expense will 
be made to keep it below a rate where the resistance is twice 
(or possibly three times) the resistance on the ruling grade, and 
then the trains can be hauled unbroken up these few special 
grades with the help of one (or two) pusher engines. So far 
as limitation of train length is concerned, these pusher grades 
are no worse than the regular ruling grades and, except for the 
expense of operating the pusher engines (which is a separate 
matter), they are not appreciably more expensive than any 
ruling grade. As before stated, the engineer cannot alter very 
greatly the ruling grade of the road when the general route has 
been decided on. He may remove sags or humps, or he may 
lower the natural grade of the route by development in order 
to bring the grade wdthin the adopted limit of ruling grade. 



§ 442. GRADE. 515 

The financial value of removing sags and humps has been con- 
sidered. It noAv remains to determine the financial relation 
between the lowest permissible ruling grade and the money 
which may profitably be spent to secure it. 

441. Choice of ruling grade. It is of course impracticable for 
an engine to drop off or pick up cars according to the grades 
which may be encountered along the line. A train load is made 
up at one terminus of a division and must run to the other 
terminus. Excluding from consideration any short but steep 
grades which may always be operated by momentum, and also 
all pusher grades, the maximum grade on that division is the 
ruling grade. 

It will evidently be economy to reduce the few grades which 
naturally would be a little higher than the great majority of 
others until such a large amount of grade is at some uniform 
limit that a reduction at all these places would cost more than 
it is worth. The precise determination of this limit is prac- 
tically impossible, but an approximate value may be at once 
determined from a general survey of the route. The distance 
apart of the termini of the division into their difference of ele- 
vation is a first trial figure for the rate of the grade. If a grade 
even approximately uniform is impossible owing to the eleva- 
tions of predetermined intermediate points, the worst place 
may be selected and the natural grade of that part of the route 
determined. If this grade is much steeper than the general 
run of the natural grades, it may be policy to reduce it by devel- 
opment or to boldly plan to operate that place as a pusher 
grade. The choice of possible grades thus has large limita- 
tions, and it justifies very close study to determine the best 
combination of grades and pusher grades. When the choice 
has narrowed down to two limits, the lower of which may be 
obtained by the expenditure of a definite extra sum, the choice 
may be readily computed, as mil be developed. 

442. Maximum train load on any grade. The tractive power 
of a locomotive has been discussed in Chap. XV, § 322. The 
net train load which may be placed behind any engine is the 
difference between the weight of the engine itself and the gross 
load which can be handled under the given circumstances, with 
a given weight on the drivers. Since the design of locomotives 
is so variable, it is impracticable to show in tabular form the 
power of all kinds of locomotives on all grades. In Table XXV 



516 



RAILROAD CONSTRUCTION. 



§442. 



are given the tractive powers of locomotives of a wide range of 
types and weights and with various ratios of adhesion. They 
may be accepted as typical figures and will serve to compute the 
effect of variations of grade on train load. In Table XXVI is 
given the total train resistance in pounds per ton for various grades 
and for various values of track resistances. By a combination 

TABLE XXV. — TRACTIVE POWER OF VARIOUS TYPES OF STANDARD- 
GAUGE LOCOMOTIVES AT VARIOUS RATES OF ADHESION. 



Type of locomotive. 


Total weight 

of engine 
and tender. 


Weight 

of 

engine 

only. 


Weight 
on the 
drivers. 


Tractive power when 

ratio of adhesion 

is 




Lbs. 


Tons. 


i 


,^ 


I 


Atlantic, 4-4-2 . . . ' 340,000 
Atlantic, 4-4-2, four 1 
cylinder compound 368,800 

Pacific, 4-6-2 1343,600 

Pacific, 4-6-2 1 403.780 

Ten-wheel, 4-6-0. . .1 321,000 

Prairie, 2-6-2 1 366,500 

Consolidation, 2-8-0; 214.000 
Consolidation, 2-8-0 1 366,700 

Mikado, 2-8-2 { 405,500 

1 


170.0 

184.4 
171.8 
201.9 
160.5 
183.2 
1 07 . 
183.3 
202.7 


199,400 

206,000 
218,000 
226,700 
201,000 
212,500 
120,000 
221,500 
259,000 


105,540 

115,000 
142,000 
151,900 
154,000 
154,000 
106,000 
197,500 
196,000 


26,385 

28,750 
35,500 
37,975 
38,500 
38,500 
26,500 
49,375 
40.000 


23,740 

25,875 
31,950 
34,180 
34,650 
34,650 
23.850 
44,440 
44,100 


21,100 

23,000 
28,400 
30,380 
30,800 
30,800 
21.200 
39,500 
39,200 



of these two tables the net train load on any grade under given 
conditions may be quickly determined For example, an 
ordinary consolidation engine having a weight of 106000 
pounds on the drivers (see Table XXV) will have a tractive 
force of 26500 pounds under fair conditions of track, when the 
adhesion ratio is J. When climbing slowly up a. grade of 1.30% 
the tractive resistance will be about 32 pounds per ton if the roll- 
ing-stock and track are fair — assuming a tractive resistance on 
a level of 6 pounds per ton. Dividing 26500 by 32 we have 
828 tons, the gross train load. Subtracting 107 tons, the weight 
of the engine and tender in working order, we have 721 tons, 
the net load. Incidentally we may note that, cutting do^\Ti 
the grade to 0.90% (a reduction of only 21.12 feet per mile), 
the resistance per ton is reduced to 24 pounds and the gross 
train load is increased to 1104 tons and the net load to 997 
tons — an increase of about 38%. 

As another numerical example, consider a contractor's loco- 
motive (not referred to in Table XXV), a light four-wheel-con- 
nected-tank narrow-gauge engine, with a total weight of 12000 
pounds, all on the drivers. On the rough temporar^^ track 
used by contractors the tractive ratio may be as low as |. 
The tractive adhesion should therefore be taken as 2400 pounds. 
Assume that the grade when hauling ^^ empties'' is 4.7% and 



§ 444. GRADE. 517 

that the tractive resistance on such a track on a level is 10 pounds 
per ton. By Table XXYI, the total train resistance is therefore 
(by interpolation) 104 pounds per ton. 2400-^-104 =23 tons; 
subtracting the weight of the engine we have 17 tons, the net 
load of empty cars — perhaps twenty cars weighing 1700 pounds 
per car. 

In general, and to compute accurately the train load under 
conditions not exactly given in the tables, the maximum train 
load ma}^ be computed according to the following rule: 

The maximum load behind an engine on any grade may be 
found by multiplying the weight on the drivers b}^ the ratio of 
adhesion and dividing this by the sum of the grade and tractive 
resistances per ton; this gives the gross load, from which the 
weight of the engine and tender must be subtracted to find the 
net load. 

443. Proportion of the traffic affected by the ruling grade. 
Some very light traffic roads are not so fortunate as to have 
a traffic which will be largely affected by the rate of the ruling 
grade. When passenger traffic is light, and when, for the sake 
of encouraging traffic, more frequent trains are run than are 
required from the standpoint of engine capacity, it may happen 
that no passenger trains are really limited by any grade on the 
road — i.e., an extra passenger car could be added if needed. 
The maximum grade then has no worse effect (for passenger 
trains) than to cause a harmless reduction of speed at a few points. 
The local freight business is frequently affected in practically 
the same way. All coal, mineral, or timber roads are affected 
by the rate of ruling grade as far as such traffic is concerned. 
Likewise the through business in general merchandise, especially 
of the heavy traffic roads, will generally be affected by the rate 
of ruling grade. Therefore in computing the effect of ruling 
grade, the total number of trains on the road should not ordi- 
narily be considered, but only the trains to Avhich cars are added, 
until the limit of the hauling power of the engine on the ruling 
grades is reached. 

444. Financial value of increasing the train load. The gross 
receipts for transporting a given amount of freight is a definite 
sum regardless of the number of train loads. The cost of a 
train mile is practically constant. If it were exactly so, the 
sa\dng in operating expenses would be strictly proportional 
to the number of trains saved. How will the cost per traiA 



518 

TABLE XXVI, 



RAILROAD CONSTRUCTION. 



§ 444. 



, total train resistance per ton (of 2000 

pounds) on various grades. 







When tractive 


re- 






When tractive 


re- 


Grade. 


sistance on a level 


Grade. 


sistance on 


a level 






in pounds per ton is 






in pounds per ton is 


Rate 


Feet 












Rate 


Feet 












per 


per 


6 


7 


8 


9 


10 


per 


per 


6 


7 


8 


9 


10 


cent. 


mile. 








9 


10 


cent. 
2.00 


mile. 
105.60 


46 


47 


48 


49 




0.00 


0.00 


6 


7 


8 


50 


.05 


2.64 


7 


8 


9 


10 


11 


.05 


108.24 


47 


48 


49 


50 


51 


.10 


5.28 


8 


9 


10 


11 


12 


.10 


110.88 


48 


49 


50 


51 


52 


.15 


7.92 


9 


10 


11 


12 


13 


.15 


113.52 


49 


50 


51 


52 


53 


.20 


10.56 


10 


11 


12 


13 


14 


.20 


116.16 


50 


51 


52 


53 


54 


0.25 


13.20 


11 


12 


13 


14 
15 


15 
16 


2.25 


118.80 


51 


52 


53 


54 


55 


.30 


15.84 


12 


13 


14 


.30 


121.44 


52 


53 


54 


~55 


56 


.35 


18.48 


13 


14 


15 


16 


17 


.35 


124.08 


53 


54 


55 


56 


57 


.40 


21.12 


14 


15 


16 


17 


18 


.40 


126.72 


54 


55 


56 


57 


58 


.45 


23.76 


15 


16 


17 


18 


19 


.45 


129.36 


55 


56 


57 


58 


59 


0.50 


26.40 


16 


17 


18 


19 


20 


2.50 


132.00 


56 


57 


58 


59 


60 


.55 


29.04 


17 


18 


19 


20 


21 


.55 


134.64 


57 


58 


59 


60 


61 


.60 


31.68 


IS 


19 


20 


21 


22 


.60 


137.28 


58 


59 


60 


61 


62 


.65 


34.32 


19 


20 


21 


22 


23 


.65 


139.92 


59 


60 


61 


62 


63 


.70 


36.96 


20 


21 


99 


23 


24 


.70 


142.56 


60 


61 


62 


63 


64 


0.75 


39.60 


21 


22 


23 


24 


^l 


2.75 


145.20 


61 


62 


63 


64 


65 


.80 


42.24 


22 


23 


24 


25 


26 


.80 


'147.84 


62 


63 


64 


65 


66 


.85 


44.88 


23 


24 


25 


26 


27 


.85 


150.48 


63 


64 


65 


66 


67 


.90 


47.52 


24 


25 


26 


27 


28 


.90 


153.12 


64 


65 


66 


67 


68 


0.95 


50.16 


25 


26 


27 


28 


29 


.95 


155.76 


65 


66 


67 


68 


69 


1.00 


52.80 


26 


27 

28 


28 
29 


29 
30 


30 
31 


3.00 


158.40 


66 


67 


68 


69 


70 


.05 


55.44 


27 


.05 


161.04 


67 


68 


69 


7C 


71 


.10 


58.08 


28 


29 


30 


31 


32 


.10 


163.68 


68 


69 


70 


71 


72 


.15 


60.72 


29 


30 


31 


32 


33 


.15 


166.32 


69 


70 


71 


72 


73 


.20 


63.36 


30 


31 


32 


33 


34 


.20 


168.96 


70 


71 


72 


73 


74 


1.25 


66.00 
68.64 


31 
32 


32 
33 


33 
34 


34 
35 


35 
36 


3.25 


171.60 


71 


72 


73 


74 


75 


.30 


.30 


174.24 


72 


73 


74 


75 


76 


.35 


71.28 


33 


34 


35 


36 


37 


.35 


176.88 


73 


74 


75 


76 


77 


.40 


73.92 


34 


35 


36 


37 


38 


.40 


179.52 


74 


75 


76 


77 


78 


.45 


76.56 


35 


36 


37 


38 


39 


.45 


182.16 


75 


76 


77 


78 


79 


1.50 


79.20 


36 


37 


38 


39 


40 


3.50 


184.80 


76 


77 


78 


79 


80 


.55 


81.84 


37 


38 


39 


40 


41 


4.00 


211.20 


86 


87 


88 


89 


90 


.60 


84.48 


38 


39 


40 


41 


42 


4.50 


237.60 


96 


97 


98 


99 


100 


.65 


87.12 


39 


40 


41 


42 


43 


5.00 


264.00 


106 


107 


108 


109 


110 


.70 


89.76 


40 


41 


42 


43 


44 


5.50 


290.40 


116 


117 


118 


119 


120 


1.75 


92.40 


41 


42 


43 


44 


45 


6.00 


316.80 


126 


127 


128 


129 


130 


.80 


95.04 


42 


43 


44 


45 


46 


6.50 


343.20 


136 


137 


138 


139 


140 


.85 


97.68 


43 


44 


45 


46 


47 


7.00 


.369.60 


146 


147 


148 


149 


150 


.90 


100 . 32 


44 


45 


46 


47 


48 


8.00 


422.40 


166 


167 


168 


169 


170 


1.95 


102.96 


45 


46 


47 


48 


49 


9.00 


475 . 20 


186 


187 


188 


189 


190 


2.00 


105.60 


46 


47 


48 


49 


50 


10.00 


528.00 


206 


207 


208 


209 


210 



mile vary when by a reduction in ruling grade more cars are 
handled in one train than before ? First^ compute the effect 



§ 444. GRADE. 519 

of increasing the train load so that one less engine will handle 
the trafhc, or, for example, that an engine can haul 11 cars 
instead of 10 or 44 instead of 40 — that 10 engines will do the 
work for which 11 engines would be required with the steeper 
grade. What will be the relative cost of running 10 heavy 
trains rather than 11 lighter trains, or, rather, what will be the 
extra cost of the extra engine ? 

Since the gross traffic to be handled is assumed to be the 
same, the number of cars required to handle it will also be the 
same whatever the number of trains, and the effect of those 
cars on the wear and tear of track, etc., T\dll e\ddently be constant. 
The locomotive, on account of the greater concentration of 
loading of the driver wheels, damages the track (in proportion 
to its tonnage) much more than the cars. It has been estimated 
that the locomotive is responsible for one half of the track 
wear Such an estimate is verified by the wear of rails on steep 
tracks around coal-mines where standard cars are hauled by 
cables. If we assume that 50% of Items 2 and 3 and of that 
part of Item 1 which varies with tonnage 'is due to the locomo- 
tives, then the extra expense caused by the extra engine w^ill 
be 50% of Items 2 and 3 and 50% of 25%, of Item 1. The 
other items of maintenance of way are unaffected except that 
truss bridges, trestles, and the maintenance of a few buildings 
will be slightly affected by the extra locomotive. But the 
actual effect is quite indefinite and is evidently very small. 

Maintenance of equipment: Engine repairs will evidently be 
affected according to the mileage. Throughout the ruling 
grade of the road (by whichever system of grades) the engines 
(assumed of uniform style) are working at their utmost capacity. 
On the lighter grades and level sections the engines will have 
easier work when the cars are fewer and this will have a tendency 
to reduce engine repairs. Suppose that by decreasing the 
number of cars 10% on the easy grades the engine repairs on 
each engine are reduced 2%. There is little or no justification 
for estimating the reduction to be more than this. Then on 
the ten engines the saving is 20% of the average charge for 1 
engine. , Suppose that by decreasing the number of cars 20% 
on the easy grades the engine repairs are reduced 4%, on the 
five engines they are reduced 20% again. In either case the 
net added cost due to the extra engine would be but 80% of 
the average cost While the above estimate is but a guess. 



520 RAILROAD CONSTRUCTION. § 445. 

yet it is very evident that the extra cost for tliis item is but 
httle less than the normal charge. 

Car repairs will be reduced by a decrease in the number of 
cars per train. The average draw-bar pull will be less, the 
wear and tear due to stoppage and starting will be less. This 
is the one item in which an increased number of trains for the 
same tonnage is an actual advantage. The saving per car is 
evidently greater when 4 trains are increased to 5 than when 
10 trains are increased to 11 ; but the saving per train added 
on is constant. Wellington estimates the saving to be 10%. 
His basis of calculation is somewhat different, but it reduces 
to the same thing. The estimate applies chiefly to Item 14 
and to Item 13 in so far as •passenger trains are affected by 
ruling grade. The other items of maintenance of equipment are 
but little, if any, affected. 

Conducting transportation. Items 21, 26, 27, 28, 29, 30, 
31, 32, 34, 35, 36, 37, 45, and 46 may be considered as varying 
according to the train mileage. While some of them seem to 
have but little direct connection with train mileage, yet if a 
road increases its traffic from 10 trains a day to 20 trains a day 
all of these items seem to increase in due proportion. 

Fuel, etc., for locomotives (Items 22-25) will increase nearly 
as the engine mileage. In either case the engines work to the 
limit of their capacity on the ruling grades. In either case the 
loss of heat due to radiation is the same. But the engines with 
the lighter trains work a little easier on the light or level grades. 
By the same course of reasoning as was given regarding engine 
repairs the fuel saving from the normal requirement for the 
extra engine vdW be about the same no matter whether there 
is an addition of one engine in 5 or 10. The saving in fuel will 
be assumed at 20% of the normal consumption, or rather that 
the use of the extra engine adds 80% of the normal charge for 
fuel. The same estimate applies to items 23, 24, and 25. 

Car mileage, item 33, is unaffected. Items 20 and 38 to 44 
will be considered as unaffected; also the general expenses. 

445. Operating value of a reduction in the rate of the ruling 
grade. Collecting the above estimates, we have Table XXVII. 
To this must be added something for the capital cost of the extra 
engine. Assume that it costs $10,000 and that its mileage life 
is 800,000 miles. This makes an average charge of 1.25 c. per 
mile. Of course the cost of operation, maintenance, and repairs 



§445. 



GRADE, 



521 



is included in the tabulated expense. 53.25% of 135 c. =71.89 c. 
Adding 1.25 c., we have 73.14. 



TABLE XXVI 



-COST OF AN ADDITIONAL TRAIN TO HANDLE 
A GIVEN TRAFFIC. 



No. 


Item (abbreviated). 


Normal 
average. 


Per cent 
affected. 


Cost per 
cent. 


1 


Roadway 


10.782 
1.403 
2.868 
5 . 943 


12.5 

50 
50 



1.35 


2 


Rails 


70 


3 


Ties 


1.43 


4-10 


Bridges, buildings, etc 







Maintenance of way 


20 . 996 




3 48 










11 


Superintendence 


.600 
7.011 
2.125 
7.561 
1.628 




80 

- 5 

-10 







12 

13 

14 

15-19 


Repairs of locomotives 

Repairs of passenger cars 

Repairs of freight cars 

Miscellaneous 


5.61 

- .11 

- .76 












Maintenance of equipment .... 


18.925 




4.74 








20 


Superintendence 


1.772 
9.527 
11.788 
22.529 
1.657 
2.444 
5.131 
1.098 




100 

80 

100 


100 


100 





21 


Enginemen 


9 53 


22-25 


Fuel, etc 


9 43 


26-32 


Train service etc 


22 5^> 


33 


Car mileage 





34-37 


Damages etc 


2 44 


38-44 


Miscellaneous 





45-46 


Stationerv, etc 


1.10 










Conducting transportation . . . 


.'^5 Q4fi 




45 03 










47-53 


General expenses 


4.133 










100.000 




53.25 



As a practical application of the above figures, assume that 
on a constructed and operated road the ruling grade on a 100- 
mile division is 1.6%; the actual traffic affected by ruling grade 
is 8 daily trains with a net load of 552 tons or 4416 tons. It 
is found that with an expenditure of $400000 the ruling grade 
may be reduced to 1.2%. Will it pay? At 1.2% grade the net 
load behind an 80-ton consolidation engine, with 48 tons on 
the drivers, adhesion J, and 6 pounds per ton normal resistance, 
is 720 tons. The traffic (4416 tons) may therefore be hauled 
by 6 engines, the balance, less than 100 tons, being taken care 
of by lighter trains not affected by the ruling grade. There is 
therefore the saving due to not operating two engines. Since 
the additional cost of the two engines drawing lighter trains is 
73.14 c. per mile, the annual saving is therefore 2 X $0.7314X100 
X 365 = $53392.20, which capitalized at 5% = $1,067,844. This 



522 RAILROAD CONSTRUCTION. § 445. 

shows that if the improvement can be accomplished for $400000 
it is worth while- 

As in other similar problems, it must be reiterated that al- 
though there are some more or less uncertain elements in the 
above estimates, yet with a considerable margin for error in 
individual items the value of the whole improvement will not 
be very greatly altered and the estimate will be infinitely better 
than an indefinite reliance on vague '^ judgment." Of course 
certain items in the above estimates are somewhat variable 
and should be altered to fit the particular case to be computed. 

PUSHER GRADES. 

446. General principles underlying the use of pusher engines. 

On nearly all roads there are some grades which are greatly 
in excess of the general average rate of grade and these heavy- 
grades cannot usually be materially reduced without an ex- 
penditure which is excessive and beyond the financial capacity 
of the road. If no pusher engines are used, the length of all 
heavy trains is limited by these grades. The financial value 
of the reduction of such ruling grades has already been shown. 
But in the operation of pusher grades there is incurred the 
additional cost of pusher-engine ser\ace, for a pusher engine 
must run twice over the grade for each train which is assisted. 
It is possible for this additional expense to equal or even exceed 
the advantage to be gained. In any case it means the adoption 
of the lesser of two evils, or the adoption of the more economical 
method. A simple example will illustrate the point. Assume 
that at one point on the road there is a grade of 1.9% which 
is five miles long. Assume that all other grades are less than 
0.92%. I^' pushers are not to be used the net capacit}^ of a 
107-ton consolidation engine with 53 tons on the drivers, assum- 
ing ^^0 adhesion and 6 pounds per ton for normal resistance, 
will be 435 tons, and that will be the maximum weight of train 
allowable. By using pusher engines on this one 5-mile grade 
the train load is at once doubled and the number of trains 
cut dowTL one half. This double load, 870 tons, can easily be 
hauled by one engine up the 92% grades. As a rough com- 
parison, free from details and allowances, we may say: 

(a) 10 trains per day over a 100-mile division, 435 tons 
net per train, will require 1000 engine miles daily. 



§ 447. GRADE. 523 

(b) 5 trains per day handling the same traffic, 870 tons 
net per train, with 2X5X5 pusher-engine miles, will require 
(5X100) + (2X5X5) =550 engine miles daily. There is thus 
a large saving in the number of engine miles and also in the 
number of the engines required for the work. Moreover, the 
engines are working to the limit of their capacity for a much 
larger proportion of the time, and their work is therefore more 
economically done. The work of overcoming the normal 
resistances of so many loaded cars over so many miles of track 
and of lifting so many tons up the gross differences of elevation 
of predetermined points of the line is approximately the same 
whatever the exact route, and if the grades are so made that 
fewer engines working more constantly can accomplish the 
work as well as more engines which are not hard worked for a 
considerable proportion of the time, the economy is very ap- 
parent and unquestionable. Wellington expresses it concisely: 
^'It is a truth of the first importance that the objection to 
high gradients is not the work which the engines have to do 
on them, but it is the work which they do not do when they 
thunder over the track with a light train behind them, from 
end to end of a division, in order that the needed power may 
be at hand at a few scattered points where alone it is needed." 

447. Balance of grades for pusher service. In the above 
illustration the ''through" grade and the ''pusher" grade are 
"balanced'' for the use of one equal pusher. It is therefore 
evident that if some intermediate grade (such as 1.4%) were 
permitted, it could only be operated b}^ (a) making it the ruling 
grade and cutting down all train loads from 870 tons to 594 tons, 
or (b) operating it as a pusher grade, although with a loss of 
economy, since two engines w^ould have much more power than/ 
necessary. The proper plan i*n such a case would be to strive 
to reduce the 1.4% grade to 0.92%, or, if that seemed imprac- 
ticable, to attempt to get an operating advantage at the expense 
of an increase of the 1.4% grade to anything short of 1.9%. 
For the increase in rate of grade would cost almost nothing, and 
some advantage might be obtained which would practically 
compensate for the introduction of a pusher grade. Another 
possible solution would be to operate the 1 9% with two pushers, 
adopt a corresponding grade for use with one pusher and a 
corresponding ruling grade for through trains With the above 



524 



RAILROAD CONSTRUCTION 



§447. 



data these three grades would be 1.90% 1.27%, and 0.54%, 
obtained as follows: 

Tractive power of three engines = 106000 x ^V X 3 = 71550 
pounds. 

Resistance on 1.9% grade= 6 + (20X 1.9) =44 lbs. per ton. 

71550-^-44= 1626 =gross load in tons. 

1626 - (3 X 107) = 1305 = net load in tons. 

1305 + (2x107) =1519 =gross load on the one-pusher grade. 

Tractive power of two engines = 47700 lbs. 

47700 H- 1519= 31.40= possible tractive force in lbs. per ton. 

(31.40 — 6) -^ 20= 1.27% = permissible grade for one pusher. 

1305 + 107= 1412 =gross load on the through grade. 

Tractive power of one engine =23850 lbs. 

23850 -=-1412 =16.89= possible tractive force in lbs. per ton. 

(16.89 — 6) -^20 =0.54% = permissible through grade. 

It should be realized that, assuming the accuracy of the 
normal resistance (6 lbs.) and the normal adhesion (^^^^j) and 
with the use of 107-ton locomotives with 53 tons on the drivers, 
the above figures are precisely what is required for hauling 
with one, two, and three engines. Other types of engines, other 
values for resistance and adhesion will vary considerably the 
gross load in tons which may be hauled up those grades, but 
starting with 0.54% as a through grade, the corresponding 
values for one and for two pushers would vary but slightly 
from those given. To show the tendency of these variations, 
the corresponding values have been computed as follows: 



Adhesion. 


Resistance 
per ton. 


Load on 
drivers. 


Through 
grade. 


One-pusher 
grade. 


Two-pusher 
grade. 




6 lbs. 

7 " 
6 " 

6 " 

7 *' 


53 tons. 
53 " 
53 •• 
53 " 
53 - 


0.54% 
.54% 
.54% 

■Ml 


1.27% 
1.31% 
1.28% 
1.26% 
1.29% 


1.90% 
1.96% 
1.93% 
1.86% 
1.92% 



The above form shows that increasing the resistance per ton 
and decreasing the adhesion have opposite effects on altering 
the ratio of these grades, and as a storm, for example, would 
increase the resistance and decrease the adhesion, the changes 
in the ratio would be compensating although the absolute 
reduction in train load might be considerable. 



§ 448. GRADE. 525 

In Table XXVIII is shown a series of "balanced'' grades on 
which a given net train load may be operated by means of one 
or two pusher engines. For example, assuming a track resistance 
of 6 pounds per ton, a consolidation engine of the type shown 
in the table can haul a train weighing 977 tons (exclusive of 
the engine) up a grade of 0.80%. If this is the maximum 
through grade, pusher grades as high as 1.70% for one pusher, 
or 2.46% for two pushers, may be introduced and the same 
net load may be hauled up these grades. 

The ratios of pusher grade to through grade, as given in 
Table XXVIII, are exactly true only for the conditions named 
as to weight and type of engine, ratio of adhesion, and norma 
track resistance. But a little comparative study of the two 
halves of Table XXVIII and of the tabular form given on page 
483 will show that although the net load which can be hauled 
on any grade varies considerably with the normal track re- 
sistance and also with the ratio of adhesion, yet the ratios of 
through to pusher grade, for either one or two pushers, varies 
but slightly with ordinary changes in these conditions. There- 
fore when the precise conditions are unknown or variable, the 
figures of Table XXVIII may be considered as applicable to 
any ordinary practice, especially for preliminary computations. 
For final calculations on any proposed ruling grade and pusher 
grade, the whole problem should be worked out on the principles 
outlined above and on the basis of the best data obtainable. 

Problem: If the through ruling grade for the road has been 
established at 1.12%, what pusher grades are permissible? 
Answer: Interpolating in Table XXVIII, we may employ a 
grade of 2.22% if the track and road-bed are to be such that a 
tractive resistance of 6 pounds per ton can be expected. With 
a poorer track, the normal resistance assumed as 8 pounds per 
ton, the rate is raised to 2.27%. The increase in rate of pusher 
grade with increase of resistance is due to the fact that the 
net load hauled is less — so much less that on the pusher grade 
a larger part of the adhesion is available to overcome a grade 
resistance. 

448. Operation of pusher engines. The maximum efficiency 
in operating pusher engines is obtained when the pusher engine 
is kept constantly at work, and this is facilitated when the pusher 
grade is as long as possible, i.e., when the heavy grades and the 
great bulk of the difference of elevation to be surmounted is 



526 



RAILROAD CONSTRUCTION. 



§448. 



TABLE XXVIII. — BALANCED GRADES FOR ONE, TWO, AND 
THREE ENGINES. 
Basis. — Through and pusher engines alike; consolidation type; total 
weight, 107 tons; weight on drivers, 53 tons; adhesion, j*^, giving a trac- 
tive force for each engine of 23850 lbs.; normal track resistance, 6 (also 8) 
lbs. per ton. 





Track resistance, 6 lbs. 


Track resistance, 


8 lbs. 






Corresponding 




Corresponding 


Through 


Net load 


pusher grade for 


Net load 


pusher grade for 


grade. 


for one 


same net load. 


for one 


same net load. 




engine m 
tons (2000 






engine in 
tons (2000 


















lbs.). 


One 


Two 


lbs«). 


One 


Two 






pusher. 


pushers. 




pusher. 


pushers. 


Level. 


3868 tons 


0.28% 


0.55% 


2874 tons 


0.37% 


0.72% 


0.10% 


2874 ' ' 


0.47% 


0.82% 


2278 ** 


0.56% 


0.98% 


0.20% 


2278 *' 


0.66% 


1.08% 


1880 " 


0.74% 


1.23% 


0.30% 


1880 " 


0.84% 


1.33% 


1596 *• 


0.92% 


1.47% 


0.40% 


1596 " 


1.02% 


1.57% 


1384 " 


1.09% 


1.70% 


0.50% 


1384 " 


1.19% 


1.80% 


1218 " 


1.27% 


1.92% 


0.60% 


1218 " 


1.37% 


2.02% 


1085 • ' 


1.44% 


2.14% 


0.70% 


1085 ' ' 


1.54% 


2.24% 


977 " 


1.60% 


2.36% 


0.80% 


977 " 


1.70% 


2.46% 


887 '• 


1.77% 


2.56% 


0.90% 


887 " 


1.87% 


2.66% 


810 " 


1.93% 


2.76% 


1.00% 


810 " 


2.03% 


2.86% 


745 " 


2.09% 


2.96% 


1 . 10% 


745 " 


2.19% 


3.06% 


688 " 


2.24% 


3.15% 


1.20% 


688 " 


2.34% 


3.25% 


638 " 


2.40% 


3.33% 


1.30% 


638 " 


2.50% 


3.43% 


594 " 


2.55% 


3.51% 


1.40% 


594 •• 


2.65% 


3.61% 


555 '• 


2.70% 


3.68% 


1.50% 


555 " 


2.80% 


3.78% 


521 •* 


2.85% 


3.85% 


1.60% 


521 " 


2.95% 


3.95% 


489 " 


2.99% 


4 . 02%, 


1.70% 


489 ♦* 


3.09% 


4.12% 


461 " 


3.13% 


4.17% 


1.80% 


461 •• 


3.23% 


4.27% 


435 " 


3.27% 


4.33% 


1.90% 


435 •• 


3.37% 


4.43% 


411 " 


3.42% 


4.49% 


2.00% 


411 " 


3.52% 


4.59% 


390 '* 


3.55% 


4.63% 

4.78% 


2.10% 


390 " 


3.65% 


4.73% 


370 " 


3.68% 


2 . 20%, 


370 " 


3.78% 


4.88% 


352 " 


3.81% 


4.92% 


2.30% 


352 " 


3.91% 


5.02% 


335 " 


3.94% 


5.05% 


2.40% 


335 " 


4.04% 


5.15% 


319 *' 


4.07% 


5.19% 


2.50% 


319 ** 


4.17% 


5.29% 


304 •* 


4.20% 


5.32% 



at one place. For example, a pusher grade of three miles fol- 
lowed by a comparatively level stretch of three miles and then 
by another pusher grade of two miles cannot all be operated as^ 
cheaply as a continuous pusher grade of five miles. Either i 
the two grades must be operated as a continuous grade of eight 
miles (sixteen pusher miles per trip) or else as two short pusher 
grades, in which case there would be a very great loss of time' 
and a difficulty in so arranging the schedules that a train need 



§ 449. GRADE. 527 

not wait for a pusher or the pushers need not waste too much 
time in idleness waiting for trains. If the level stretch were 
imperative, the two grades would probably be operated as one, 
but an effort should be made to bring the grades together. It 
is not necessary to bring the trains to a stop to uncouple the 
pusher engine, but a stop is generally made for coupling on, and 
the actual cost in loss of energy and in wear and tear of stopping 
and starting a heavy train is as great as the cost of running 
an engine light for several miles. 

There are two ways in which it is possible to economize in 
the use of pusher engines, (a) When the traffic of a road is 
so very light that a pusher engine will not be kept reasonably 
busy on the pusher grade it may be worth while to place a 
siding long enough for the longest trains both at top and bottom 
of the pusher grade and then take up the train in sections. 
Perhaps the worst objection to this method is the time lost 
while the engine runs the extra mileage, but with such very 
light traffic roads a little time more or less is of small consequence. 
On light traffic roads this method of surmounting a heavy grade 
will be occasionally adopted even if pushers are never used. 
If the traffic is fluctuating, the method has the advantage 
of only requiring such operation when it is needed and avoiding 
the purchase and operation of a pusher engine which has but 
little to do and which might be idle for a considerable proportion 
of the year, (b) The second possible method of economizing 
is only practicable when a pusher grade begins or ends at or 
near a station yard where switching-engines are required. In 
such cases there is a possible economy in utilizing the switching- 
engines as pushers, especially when the work in each class is 
small, and thus obtain a greater useful mileage. But such cases 
are special and generally imply small traffic. 

A telegraph-station at top and bottom of a pusher grade is 
generally indispensable to effective and safe operation. 

449. Length of a pusher grade. The virtual length of the 
pusher grade, as indicated by the mileage of the pusher engine, 
is always somewhat in excess of the true length of the grade 
as shown on the profile, and sometimes the excess length is 
very great. If a station is located on a lower grade within a 
mile or so of the top or bottom of a pusher grade, it will ordina- 
rily be advisable to couple or uncouple at or near the station, 
since the telegraph-station, switching, and signaling may be 



528 RAILROAD CONSTRUCTION. § 450. 

more economically operated at a regular station. If the extra 
engine is coupled on ahead of the through engine (as is some- 
times required by law for passenger trains) the uncoupling at 
the top of the grade may be accomplished by running the assist- 
ant engine ahead at greater speed after it is uncoupled, and, 
after nmning it on a siding, clearing the track for the train. 
But this requires considerable extra track at the top of the grade. 
Therefore, when estimating the length of the pusher grade, 
the most desirable position for the terminal sidings must be 
studied and the length determined accordingly rather than 
by measuring the mere length of the grade on the profile. Of 
course these odd distances are always excess; the coupling or 
uncoupling should not be done while on the grade. 

450. Cost of pusher-engine service. The cost evidently de- 
pends partly on the mileage run, while some items are wholly 
independent of the mileage. A pusher engine, when working 
on grades where the conditions are fairly favorable, will ac- 
comphsh a mileage of 100 to 125 miles per day, and this is 
about equal to that of an ordinary freight engine. Therefore 
such items as wages which are independent of mileage will be 
assumed to cost as miuch per mile as they do for ordinary train 
service. If the mileage is less than this, an extra allowance 
should be made. 

The effect of a pusher engine on maintenance of way may 
be considered to be the same as that produced by an additional 
engine, as developed in § 444. The same allowance (3.48%) 
will therefore be made. The cost of repairs and renewals of 
locomotives may be estimated the same as for other engines. 
Wages for engine and round-house men will be the same. There 
is certainly no ground for considering that the cost of fuel and 
other engine supplies can be materially less than the usual 
figures. On the return trip do^Ti the grade the engine runs 
almost without steam (after getting started), but, on the other 
hand, the engine works hard when climbing up the grade. The 
cost of switchmen, etc., and telegraph expenses (Items 28 and 
29) will evidently add their full quota. Collecting these items, 
we have 37.80% or 51.03 c. for each mile run. Adding, as in 1 
§ 445, 1.25 c. as i^tw.est charge on the cost of the engine, we 
have 52.28 c. Then each mile of the incline will cost twice 
this or 104.56 €. for a round trip, or 104.56X365 = $382 per year 
per mile of incline per daily train needing assistance. 



§451. 



GRADE. 



529 



TABLE XXIX. ITEMS OF THE COST PER MILE OF A PUSHER ENGINE. 



No. 


Items. 


Normal 
average. 


Per cent 
affected. 


Cost per 

engine 

mile, 

per cent. 


1 


Repairs of roadway 


10.782 
1.403 
2.868 
7.011 
9.527 

11.788 
4.157 
1.828 


12.5 

50 

50 
100 
100 
100 
100 
100 


1.35 


2 


Renewals of rails 


.70 


3 


Renewals of ties 


1.43 


12 

21 


Repairs of locomotives 

Enginemen 


7.01 
9.53 


22-25 


Engine supplies 


11.79 


28 


Switchmen, etc 


4.16 


29 


Telegraph 


1.83 














37.80 



451. Numerical comparison of pusher and through grades. 
In § 445 the computation was made of the desirability of re- 
ducing a 1.6% ruling grade to a 1.2% grade. Suppose it is 
found that by keeping the 1.6% grades as pusher grades having 
a total length of 20 miles on a 100 mile division, the other grades 
may be reduced to a grade not exceeding 0.713% (the correspond- 
ing through grade) for an expenditure of $200000. Will it 
pay? The saving by cutting down trains from 8 to 4, computed 
as before, would be (see § 445), 4 X $0.7314X100X365 = $106784. 
But this saving is only accomplished by the employment of 
pushers making four round trips over 20 miles of pusher grades 
at a cost of 4 X 20 X $382 = $30560. 

The net annual saving is therefore $76224, which when 
capitalized at 5% = $1,524,480. 

The above estimate probably has this defect. The total 
daily pusher-engine mileage is but 2X4X20=160, scarcely 
work enough for two pushers. Unless the pusher grades were 
bunched into two groups of about 10 miles each, two pusher 
engines could not do the work. If the number of trains was 
much larger, then the above method of calculation would be 
more exact even though the 20 miles of pusher grade was divided 
among four or five different grades. Therefore with the above 
data the annual cost of the pusher service would probably be 
much more — perhaps twice as much — and the annual saving 
about $45000, which would justify an expenditure of $900000. 
But even this would very amply justify the assumed expenditure 
of $200000 which would accomplish this result. 

The above computation is but an illustration 01 the general 



530 RAILROAD CONSTRUCTION. § 452. 

truth which has been previously stated. In spite of the un- 
certainties and the variations of many items in the above esti- 
mates it will generally be possible to make a computation which 
will show unquestionably, as in the above instance, what is 
the best and the most economical method of procedure. When 
the capitalized valuations of both methods are so nearly equal 
that a proper choice is more difficult, the question will frequently 
be determined by the relative ease of raising additional capital. 

BALANCE OF GRADES FOR UNEQUAL TRAFFIC. 

452. Nature of the subject. It sometimes happens, as when 
a road runs into a mountainous country for the purpose of 
hauling therefrom the natural products of lumber or minerals, 
that the heavy grades are all in one direction — that the whole 
line consists of a more or less unbroken climb having perhaps 
a few comparatively level stretches, but no down grade (except 
possibly a slight sag) in the direction of the general up grade. 
With such lines this present topic has no concern. But the 
majority of railroads have termini at nearly the same level 
(500 feet in 500 miles has no practical effect on grade) and 
consist of up and down grades in nearly equal amounts and 
rates. The general rate of ruling grade is determined by the 
character of the country and the character and financial backing 
of the road to be built. It is always possible to reduce the grade 
at some point by *' development '^ or in general by the expen- 
diture of more money. It has been tacitly assumed in the 
previous discussions that when the ruling grade has been de- 
termined all grades in either direction are cut down to that 
limit. If the traffic in both directions were the same this would 
be the proper policy and sometimes is so. But it has developed, 
especially on the great east and west trunk lines, that the weight 
of the eastbound freight traffic is enormously greater than that 
of the w^estbound — that westbound trains consist very largely of 
^'empties" and that an engine which could haul twenty loaded 
cars up a given grade in eastbound traffic could haul the same 
cars empty up a much higher grade when running west. As 
an illustration of the large disproportion which may exist, the 
eastbound ton-mileage on the P. R. R. between the years 1851 
and 1885 was 3.7 times the westbound ton-mileage. Between 
the years 1876 and 1880 the ratio rose to more than 4.5 to 1. 



I 453. GRADE. 531 

On such a basis it is as important and necessary to obtain, say, 
a 0.6% ruling grade against the eastbound traffic as to have, 
say, a 1.0% grade against the westbound traffic. This is the 
basis of the following discussion. It now remains to estimate 
the probable ratio of the traffic in the two directions and from 
that to determine the proper "balance" of the opposite ruling 
grades. 

453. Computation of the theoretical balance. Assume first, 
for simplicity, that the exact business in either direction is 
accurately known. A little thought will show the truth of the 
following statements. 

1. The locomotive and passenger-car traffic in both directions 
is equal. 

2. Except as a road may carry emigrants, the passenger 
traffic in both directions is equal. Of course there are innumer- 
able individual instances in which the return trip is made by 
another route, but it is seldom if ever that there is any marked 
tendency to uniformity in this. Considering that a car load 
of, say, 50 passengers at 150 pounds apiece weigh but 7500 
pounds, which is ^ of the 45000 pounds which the car may 
weigh, even a considerable variation in the number of passengers 
will not appreciably affect the hauling of cars on grades. On 
parlor-cars and sleepers the ratio of live load to dead load (say 
20 passengers, 3000 pounds, and the car, 75000 pounds) is 
even more insignificant. The effect of passenger traffic on 
balance of grades may therefore be disregarded. 

3. Empty cars have a greater resistance per ton than loaded 
cars. Therefore in computing the hauling capacity of a loco- 
motive hauling so many tons of "empties," a larger figure must 
be used for the ordinary tractive resistances — say four pounds 
per ton greater. 

4. Owing to greater or less imperfections of management a 
small percentage of cars will run empty or but partly full in 
the direction of greatest traffic. 

5. Freight having great bulk and weight (such as grain, 
lumber, coal, etc.) is run from the rural districts toward the 
cities and manufacturing districts. 

6. The return traffic — manufactured products — although worth 
as much or more, do not weigh as much. 

As a simple numerical illustration assume that the w^eight 
of the cars is J and the live load f of the total load when 



532 RAILROAD CONSTRUCTION. § 453. 

the cars are "fuir' — although not loaded to their absolute 
limit of capacity. Assume that the relative weight of live load 
to be hauled in the other direction is but ^; assume that the 
grade against the heaviest traffic is 0.9%. Since the tractive 
resistance per ton is considerably greater in the case of unloaded 
cars than it is in the case of loaded cars, allowance must be 
made for this in calculating the train resistance. Mr. A. C. 
Dennis, of the Canadian Pacific Railway Company, has made 
some elaborate tests of train resistance for trains which were 
alternately loaded and empty, and found that the tractive resist- 
ance of loaded cars was very uniform at 4.7 pounds per ton^ 
when the weight of the empty cars was J of the total wojgM. 
He also found that the tractive resistance of empty cars was 
very uniform at 8.9 pounds per ton. Although the live load 
capacity of a box-car is usually considerably more than twice 
the weight of the empty car, it will probably coincide more 
nearly with actual running conditions to consider that the live 
load is just twice the dead load. Assume that these loads are 
being hauled by a consolidation engine with a total weight, 
including engine and tender, of 107 tons, of which 106000 pounds 
is on the drivers. We will assume that the tractive resistance 
of the locomotive is likewise 4.7 pounds per ton. On the 0.9% 
grade, the grade resistance will be 18 pounds per ton, and there- 
fore the total resistance is 22.7 pounds per ton. Assume that 
this engine is working with a tractive adhesion of J; the trac- 
tive power at the circumference of the drivers will be J of 
106000 pounds, or 26500 pounds. Dividing this by 22.7, we 
obtain 1167 as the gross load of the train in tons. Subtracting 
the weight of the locomotive, 107 tons, we have 1060 t^ns as 
the weight of the loaded cars which could be hauled by this 
locomotive up a 0.9% grade, assuming an adhesion of \. Since 
the traffic in the other direction is but ^, we will assume that 
J of the return cars are empty. We then have 353 tons of 
loaded cars with a locomotive A^eighing 107 tons, and 236 tons 
of empty cars in the return train. The loaded cars with the 
locomotive will weigh 460 tons, and their tractive resistance 
will be 4.7 pounds per ton, or 2162 pounds. The 236 tons of 
empty cars will have a resistance of 8.9 pounds per ton, or a 
total tractive resistance of 2100 pounds. This makes a total 
of 4262 pounds of tractive resistance. Subtracting this from 
the 26500 of total adhesion of the drivers, we have left 22238 



§ 453. GRADE. 533 

as the amount of pull available for grade. But the return train 
weighs 696 tons. Dividing this into 22238, we find that 32 
pounds per ton is available for grade, which is the resistance on 
a 1.60% grade. Therefore, under the above conditions, a 0.9% 
grade against the heaviest traffic will correspond with a 1.60% 
grade against the lighter traffic. 

Of course these figures will be slightly modified by variations 
in the assumptions as to the tractive resistance of loaded and 
unloaded cars, and more especially by variations in the ratio 
of live load to dead load in the two directions. Therefore no 
great accuracy can be claimed for the ratio of these tw^o grades 
in opposite directions, nevertheless the above calculation shows 
unmistakably that under the given conditions, a very consider- 
able variation in the rate of grade in opposite directions is not 
only justifiable, but a neglect to allow for it would be a great 
economic error. 

454. Computation of relative traffic. Some of the principal 
elements have already been referred to, but in addition the 
following facts should be considered. 

(a) The greatest disparity in traffic occurs through the hand- 
ling of large amounts of coal, lumber, iron ore, grain, etc. On 
roads which handle but little of these articles or on which for 
local reasons coal is hauled one way and large shipments cf 
grain the other way the disparity wall be less and will perhaps 
be insignificant. 

(h) A marked change in the development of the country iray, 
and often does, cause a marked difference in the disparity of 
traffic. The heaviest traffic (in mere w^eight) is always toward 
manufacturing regions and away from agricultural regions. But 
when a region, from being purely agricultural or mineral, be- 
comes largely manufacturing, or w^hen a manufacturing region 
develops an industry which will cause a growth of heavy freight 
traffic from it, a marked change in the relative freight movement 
will be the result. 

(c) Very great fluctuations in the relative traffic may be 
expected for prolonged intervals. 

{d) An estimate of the relative traffic may be formed by 
the same general method used in computing the total traffic 
of the road (see § 373, Chap. XIX) or by noting the relative 
traffic on existing roads w^hich may be assumed to have practically 
the same traffic as the proposed road will obtain. 



CHAPTER XXIV. 
THE IMPROVEMENT OF OLD LINES. 

455. Classification of improvements. The improvements here 
considered are only those of alignment — horizontal and vertical. 
Strictly there is no definite limit, either in kind or magnitude, 
to the improvements which may be made. But since a railroad 
cannot ordinarily obtain money, even for improvements, to 
an amount greater than some small proportion of the pre- 
viously invested capital, it becomes doubly necessary to expend 
such money to the greatest possible advantage. It has been 
previously shown that securing additional business and increas- 
ing the train load are the two most important factors in increas- 
ing dividends. After these, and of far less importance, come 
reductions of curvature, reductions of distance (frequently of 
doubtful policy, see Chap. XXI, §414), and elimination of sags 
and humps. These various improvements will be briefly dis- 
cussed. 

(a) Securing additional business. It is not often possible 
by any small modification of alignment to materially increase 
the business of a road. The cases which do occur are usually 
those in which a gross error of judgment w^as committed during 
the original construction. For instance, in the early history 
of railroad construction many roads were largely aided by the 
towns through which the road passed, part of the money neces- 
sary for construction being raised by the sale of bonds, which 
were assumed or guaranteed and subsequently paid by the 
towns. Such aid was often demanded and exacted by the 
promoters. Instances are not unknown w^here a failure to 
come to an agreement has caused the promoters to dehberately 
pass by the town at a distance of some miles, to the mutual 
disadvantage of the road and the town. If the towTi subsequent- 
ly grew in spite of this disadvantage, the annual loss of business 
might readily amount to more than the original sum in dispute. 

534 



§ 457. IMPRO'^EMENT OF OLD LINES. 535 

Such an instance would be a legitimate opportunity for study 
of the advisability of a re-location. 

As another instance (the original location being justifiable) 
a railroad might have been located along the bank of a consider- 
able river too wide to be crossed except at considerable expense. 
When originally constructed the enterprise would not justify 
the two extra bridges needed to reach the town. A growth in 
prosperity and in the business obtainable might subsequently 
make such extra expense a profitable investment. 

(b) Increasing the train load. On account of its importance 
this will be separately considered in § 458 et seq. 

(c) Reductions in curvature and distance and the elimination 
of sags and humps. The financial value of these improvements 
has already been discussed in Chapters XXI^ XXII, and XXIII. 
Such improvements are constantly being made by all progressive 
roads. The need for such changes occurs in some cases because 
the original location was very faulty, the revised location being 
no more expensive than the original, and in other cases because 
the original location was the best that was then financially 
possible and because the present expanded business will justify 
a change. 

(d) Changing the location of stations or of passing sidings. 
The station may sometimes be re-located so as to bring it nearer 
to the business center and thus increase the business done. 
But the principal reasons for re-locating stations or passing 
sidings is that starting trains may have an easier grade on which 
to overcome the additional resistances of starting. Such changes 
will be discussed in detail in § 460. 

456. Advantages of re-locations. There are certain undoubted 
advantages possessed by the engineer who is endeavoring to 
improve an old line. 

(a) The gross trafl^c to be handled is definitely known. 

(b) The actual cost per train-mile for that road (which may 
differ very greatly from the average) is also known, and therefore 
the value of the proposed improvement can be more accurately 
determined. 

(c) The actual performance of such locomotives as are used 
on the road may be studied at leisure and more reliable data 
may be obtained for the computations. 

457. Disadvantages of re-locations. The disadvantages are 
generally more apparent and frequently appear practically 



536 RAILROAD CONSTRUCTION. § 457. 

insuperable — more so than they prove to be on closer inspection, 
(a) It frequently means the abandonment of a greater or less 
length of old line and the construction of new line. At first 
thought it might seem as if a change of line such as would permit 
an increase of train-load of 50 or perhaps 100% could never 
be obtained, or at least that it could not be done except at an 
impracticable expense. On the contrary a change of 10% 
of the old line is frequenth^ all that is necessary to reduce the 
grades so that the train-loads hauled by one engine ma}^ be 
nearly if not quite doubled. And when it is considered that 
the cost of a road to sub-grade is generally not more than one- 
third of the total cost of construction and equipment per mile, 
it becomes plain that an expenditure of but a small percentage 
of the original outlay, expended where it will do the most good, 
will often suffice to increase enormously the earning capacity. 

(b) One of the most difficult matters is to convince the finan- 
cial backers of the road that the proposed improvement will 
be justifiable. The cause is simple. The disadvantages of the 
original construction lie in the large increase of certain items 
of expense which are necessary to handle a given traffic. And 
yet the fact that the expenditures are larger than they need 
be are only apparent to the expert, and the fact that a saving 
may be made is considered to be largely a matter of opinion 
until it is demonstrated by actual trial. On the other hand 
the cost of the proposed changes is definite, and the very fact 
that the road has been uneconomically worked and is in a poor 
financial condition makes it difficult to obtain money for im- 
provements. 

(c) The legal right to abandon a section of operated line 
and thus reduce the value of some adjoining property has 
sometimes been successfully attacked. A common instance 
would be that of a factory which was located adjoining the right 
of way for convenience of transportation facilities. The abandon- 
ment of that section of the right of way would probably be fatal 
to the successful operation of the factory. The objection may 
be largely eliminated by the maintenance of the old right of 
way as a long siding (although the business of the factory might 
not be worth it), but it is not alwaj^s so easy of solution, and 
this phase of the question must always be considered. 



* 458. IMPROVEMENT OF OLD LINES. 537 



REDUCTION OF VIRTUAL GRADE. 

458. Obtaining data for computations. As developed in the 
last chapter (§§ 432-434) the real object to be attained is the 
reduction of the virtual grade. The method of comparing grades 
under various assumed conditions was there discussed. A^Hien 
the road is still ''on paper'' some such method is all that is 
possible; but when the road is in actual operation the virtual 
grade of the road at various critical points, with the rolling 
stock actually in use, may be determined by a simple test and 
the effect of a proposed change may be reliably computed. 
Bearing in mind the general principle that the \drtual grade 
line is the locus of points determined by adding to the actual 
grade profile ordinates equal to the velocity head of the train, 
it only becomes necessary to measure the velocity at various 
points. Since the velocity is not usually uniform, its precise 
determination at any instant is almost impossible, but it will 
generally be found to be sufficiently precise to assume the velocity 
to be uniform for a short distance, and then observe the time 
required to pass that short space. Suppose that an ordinary 
watch is used and the time taken to the nearest second. At 
30 lailes per hour, the velocity is 44 feet per second. To obtain 
the time to within 1%, the time would need to be 100 seconds 
and the space 4400 feet. But with variable velocity there 
would be too great error in assuming the velocity as uniform 
for 4400 feet or for the time of 100 seconds. Using a stop- 
watch registering fifths of a second, a 1% accuracy would 
require but 20 seconds and a space of 880 feet, at 30 miles per 
hour. Wellington suggests that the space be made 293 feet 
4 inches, or 3-V ^^ ^ mile; then the speed in miles per hour 
equals 200-^s, in which s is the time in seconds required to 
traverse the 293' 4''. For instance, suppose the time required 
to pass the interval is 12.5 seconds. -^^ mile in 12.5 seconds = 
one mile in 225 seconds, or 16 miles per hour. But likemse 
200^12.5=16, the required velocity. The following features 
should be noted when obtaining data for the computations: 

(a) All critical grades on the road should be located and 
their profiles obtained — by a survey if necessary. 

(6) At the bottom and top of all long grades (and perhaps at 
intermediate points if the grades are very long) spaces of known 



538 RAILROAD CONSTRUCTION. § 458. 

length (preferably 293J feet) should be measured off and marked 
by flags, painted boards, or any other ser\dceable targets. 

(c) Provided with a stop-watch marking fifths of seconds 
the observer should ride on the trains affected by these grades 
and note the exact interval of time required to pass these spaces. 
If the space is 293 J feet, the velocity in miles per hour =200 -r- 
interval in seconds. In general, 

_ distance in feet X 3600 
time in seconds X 5280* 

(d) Since these critical grades are those which require the 
greatest tax on the power of the locomotive, the conditions 
under which the locomotive is working must be known — i.e., 
the steam pressure, point of cut-off, and position of the throttle. 
Economy of coal consumption as well as efficient working at 
high speeds requires that steam be used expansively (using an 
early cut-off) , and even that the throttle be partly closed ; but 
when an engine is slowly climbing up a maximum grade with a 
full load it is not exerting its maximum tractive power unless 
it has its maximum steam pressure, wide-open throttle, and is 
cutting off nearly at full stroke. These data must therefore 
be obtained so as to know whether the engine is developing 
at a critical place all the tractive force of which it is capable. 
The condition of the track (wet and slippery or dry) and the 
approximate direction and force of the mnd should be noted 
^vith sufficient accuracy to judge whether the test has been made 
under ordinary conditions rather than under conditions which 
are exceptionally favorable or unfavorable. 

(e) The train-loading should be obtained as closely as possible. 
Of course the dead weight of the cars is easily found, and the 
records of the freight department will usually give the live 
load mth all sufficient accuracy. 

459. Use of the data obtained. A very brief inspection 
of the results, freed from refined calculations or uncertainties, 
will demonstrate the following truths: 

(a) If, on a uniform grade, the velocity increases, it shows 
that, under those conditions of engine working, the load is less 
than the engine can handle on that grade 

(b) If the velocity decreases, it shows that the load is greater 
than the engine can handle on an indefinite length of such 



§ 459. IMPROVExMENT OF OLD LINES. 539 

grade. It shows that such a grade is being operated by momen- 
tum. From the rate of decrease of velocity the maximum 
practicable length of such a grade (starting with a given velocity) 
may be easily computed. 

(c) By combining results under different conditions of grade 
but with practically the same engine working, the tractive 
power of the engine may be determined (according to the prin- 
ciples previously demonstrated) for any grade and velocity. 
For example: On an examination of the profile of a division 
of a road the maximum grade was found to be 1.62% (85.54 
feet per mile). At the bottom and near the top of this grade 
two lengths of 293' 4" are laid off. The distance between the 
centers of these lengths is 6000 feet. A freight train moving 
up the grade is timed at 9f seconds on the lower stretch and 7| 

seconds on the upper. These times correspond to ^-— and ;=— ^ 

or 21.3 and 26.3 miles per hour respectively. It is at once 

observed that the velocity has increased and that the engine 

could draw even a heavier load up such a grade for an indefinite 

distance. How much heavier might the load be? 

For simpHcity we will assume that the conditions were 

normal, neither exceptionally favorable nor unfavorable, and 

that the engine was worked to its maximum capacity. The 

engine is a ^^consoHdation" weighing 128700 pounds, with 

112600 pounds on the drivers. The train-load behind the 

engine consists of ten loaded cars weighing 465 tons and eleven 

empties weighing 183 tons, thus making a total train- weight of 

712 tons. Applying Eq. 140, we find that the additional force 

which the engine has actually exerted per ton in increasing the 

velocity from 21.3 to 26.3 miles per hour in a distance of 6000 

feet is 

70 224 
P - -^^(26.32-21 .32) =2.78 pounds per ton 

The grade resistance on a 1.62% grade is 32.4 pounds per 
ton. The average train resistance may be computed similarly 
to the method adopted in § 439: 

465 1 

I tons at 4.7 pounds per ton = 2486 pounds 

183 '* '* 8.9 *' '' " =1629 '' 
.___ (( 

712 4115 '' 



540 RAILROAD CONSTRUCTION. 

The average tractive resistance is therefore 4115^712 = 5.78 
pounds per ton. Adding the grade resistance (32.4) we have 
a total train resistance of 38.18 pounds per ton. But, com- 
puting from the increase in velocity, the locomotive is evidently- 
exerting a pull of 2.78 pounds per ton in excess of the computed 
required pull on that grade, or a total pull of 40.96 pounds 
per ton. Therefore the train load might have been increased 
proportionately and might have been made 



^,^^^ 2.78 + 38.18 _ . . 
712X qqTq = 764 tons. 



This shows that 52 tons additional might have been loaded 
on to the train, or say, three more empties or one additional 
loaded car. 

A pull of 40.96 pounds per ton means a total adhesion at the 
drivers of 29164 pounds, which is about 26% of the weight on 
the drivers — 112600 pounds. This indicates average condi- 
tions as to traction, although better conditions than can be 
depended on for regular service. 

The above calculation should of course be considered simply 
as a "single observation." The performance of the same engine 
on the same grade (as well as on many other grades) on succeed- 
ing days should also be noted. It may readily happen that 
variations in the condition of the track or of the handling of the 
engine may make considerable variation in the results of the 
several calculations, but when the work is properly done it is 
always possible to draw definite and very positive deductions. 

460. Reducing the starting grade at stations. The resistance 
to starting a train is augmented from two causes : (a) the trac- 
tive resistances are usualh^ about 20 pounds per ton instead 
of, say, 6 pounds, and (6) the inertia resistance must be overcome. 
The inertia resistance of a freight train (see § 347) which is 
expected to attain a velocity of 15 miles per hour in a distance 
of 1000 feet is (see Eq. 140) 

70 224 
P = ' (15^ — 0) = 15 . 8 pounds per ton, which is the equiva- 
lent of a 0.79% grade. Adding this to a grade which nearly or 
quite equals the ruling grade, it virtually creates a new and 
higher ruling grade. Of course that additional force can be 
greatly reduced at the expense of slower acceleration, but even 



§460, 



IMPROVEMENT OF OLD LINES. 



541 



this cannot be done indefinitely, and an acceleration to only 
15 miles per hour in 1000 feet is as slow as should be allowed 
for. With perhaps 14 pounds per ton additional tractive 
resistance, we have about 30 pounds per ton additional — equiva- 




FiQ. 217. 



lent to a 1.5% grade. Instances are known where it has proven 
wise to create a hump (in what was otherwise a uniform grade) 
at a station. The effect of this on high-speed passenger trains 
moving up the grade w^ould be merely to reduce their speed 
ver}' slightly. No harm is done to trains moving down the 
grade. Freight trains moving up the grade and intending to 
stop at the station will merely have their velocity reduced as 
they approach the station and will actually save part of the 
wear and tear otherwise resulting from applying brakes. When 
the trains start they are assisted by the short down grade, 
just w^here they need assistance most. Even if the grade CD 
is still an up grade, the pull required at starting is less than that 
required on the uniform grade by an amount equal to 20 times 
the difference of the grade in per cent. 



APPENDIX. 
THE ADJUSTMENTS OF mSTETJMENTS. 

The accuracy of instrumental work may be vitiated by any 
one of a large number of inaccuracies in the geometrical relations 
of the parts of the instruments. Some of these relations are so 
apt to b*^ pltered by ordinary usage of the instrument that the 
makers have provided adjusting-screws so that the inaccuracies 
may be readily corrected. There are other possible defects, 
which, however, will seldom be found to exist, provided the 
instrument was properly made and has never been subjected to 
treatment sufficiently rough to distort it. Such defects, when 
found, can only be corrected by a competent instrument-maker 
or repairer. 

A WARNING is necessary to those who would test the accuracy 
of instruments, and especially to those whose experience in such 
work is small. Lack of skill in handling an instrument will 
often indicate an apparent error of adjustment when the real 
error is very different or perhaps non-existent. It is always a 
safe plan when testing an adjustment to note the amount of the 
apparent error; then, beginning anew, make another independent 
determination of the amount of the error. When two or more 
perfectly independent determinations of such an error are made 
it will generally be found that they differ by an appreciable 
amount. The differences may be due in variable measure to 
careless inaccurate manipulation and to instrumental defects 
which are wholly independent of the particular test being made. 
Such careful determinations of the amounts of the errors are 
generally advisable in view of the next paragraph. 

Do NOT DISTURB THE ADJUSTING-SCREWS ANY MORE THAN 

NECESSARY. Although metals are apparently rigid, they are 
really elastic and yielding. If some parts of a complicated 
mechanism, which is held together largely by friction, are sub- 
jected to greater internal stresses than other parts of the mech- 

542 



APPENDIX. 543 

anism, the jarring resulting from handling will frequently causa 
a slight readjustment in the parts which will tend to more nearly 
equalize the internal stresses. Such action frequently occurs 
with the adjusting mechanism of instruments. One screw may 
be strained more than others. The friction of parts may pre- 
vent the opposing screw from immediately taking up an equal 
stress. Perhaps the adjustment appears perfect under these 
conditions Jarring diminishes the friction between the parts, 
and the unequal stresses tend to equalize. A motion takes place 
which, although microscopically minute, is sufficient to indicate 
an error of adjustment. A readjustment made by unskillful 
hands may not make the final adjustment any more perfect. 
The frequent shifting of adjusting-screws wears them badly, 
and when the screws are worn it is still more difficult to keep 
them from moving enough to vitiate the adjustments. It is 
therefore preferable in many cases to refrain from disturbing the 
adjusting-screws, especially as the accuracy of the work done is 
not necessarily affected by errors of adjustment, as may be 
illustrated : 

(a) Certain operations are absolutely unaffected by certain, 
errors of adjustment. 

{h) Certain operations are so slightly affected by certain smAill 
errors of adjustment that their effect may properly be neglected. 

(c) Certain errors of adjustment may be readily allowed for 
and neutralized so that no error results from the use of the 
unadjusted instrument. Illustrations of all these cases will be 
given under their proper heads. 

ADJUSTMENTS OF THE TRANSIT. 

1. To have the plate-huhbles in the center of the tubes when the 
axis is vertical. Clamp the upper plate and, with the lower 
clamp loose, swing the instrument so that the plate-bubbles are 
parallel to the lines of opposite leveling-screws. Level up until 
both bubbles are central. Swing the instrument 180°. If the 
bubbles again settle at the center, the adjustment is perfect. If 
either bubble does not settle in the center, move the leveling- 
screws until the bubble is half-way back to the center. Then, 
before touching the adjusting-screws, note carefully the position 
of the bubbles and observe whether the bubbles always settle at 
the same place in the tube^ no matter to what position the in- 



544 RAILROAD CONSTRUCTION. 

strument may be rotated. When the instrument is so leveled, 
the axis is truly vertical and the discrepancies between this 
constant position of the bubbles and the centers of the tubes 
measure the errors of adjustment. By means of the adjusting- 
screws bring each bubble to the center of the tube. If this is 
done so skillfully that the true level of the instrument is not 
disturbed, the bubbles should settle in the center for all positions 
of the instrument. Under unskillful hands, two or more such 
trials ma}' be necessary. 

When the plates are not horizontal, the measured angle is greater than 
the true horizontal angle by the difference between the measured angle 
and its projection on a horizontal plane. When this angle of inclination 
is small, the difference is insignificant. Therefore when the plate-bubbles 
are very nearly in adjustment, the error of measurement of horizontal 
angles may be far within the lowest unit of measurement used. A small 
error of adjustment of the plate-bubble perpendicular to the telescope will 
affect the horizontal angles by only a small proportion of the error, which 
will be perhaps imperceptible. Vertical angles will be affected by the 
same insignificant amount. A small error of adjustment of the plate- 
bubble parallel to the telescope will affect horizontal angles very slightly, 
but will affect vertical angles by the full amount of the error. 

All error due to unadjusted plate-bubbles may be avoided by noting in 
what positions in the tubes the bubbles will remain fixed for all positions 
of azimuth and then keeping the bubbles adjusted to these positions, for 
the axis is then truly vertical. It will often save time to work in this way 
temporarily rather than to stop to make the adjustments. This should 
especially be done when accurate vertical angles are required. 

When the bubbles are truly adjusted, they should remain stationary 
regardless of whether the telescope is revolved with the upper plate loose 
and the lower plate clamped or whether the whole instrument is revolved, 
the plates being clamped together. If there is any appreciable difference, 
it shows that the two vertical axes or ''centers" of the plates are not con- 
centric. This may be due to cheap and faulty construction or to the exces- 
sive wear that may be sometimes observed in an old instrument originally 
well made. In either case it can only be corrected by a maker. 

2. To make the revolving axis of the telescope "perpendicular to 
the vertical axis of the instrument. This is best tested by using 
a long plumb-line, so placed that the telescope must be pointed 
upward at an angle of about 45° to sight at the top of the plumb- 
line and doTVTiward about the same amount, if possible, to 
sight at the lower end. The vertical axis of the transit must 
be made truly vertical. Sight at the upper part of the line^ 
clamping the horizontal plates. Swing the telescope down 
and see if the cross- wire again bisects the cord. If so, the 
adjustment is probably perfect (a conceivable exception will be 



APPENDIX. 545 

noted later) ; if not, raise or lower one end of the axis by mtang 
of the adjusting-screws, placed at the top of one of the standards, 
until the cross-wire will bisect the cord both at top and bottom. 
The plumb-bob may be steadied, if necessary, by hanging it 
in a pail of water. As many telescopes cannot be focused 
on an object nearer than 6 or 8 feet from the telescope, this 
method requires a long plumb-line swung from a high point, 
which may be inconvenient. 

Another method is to set up the instrument about 10 feet 
from a high wall. After leveling, sight at some convenient 
mark high up on the wall. Swing the telescope down and make 
a mark (when Avorking alone some convenient natural mark may 
generally be found) low down on the wall. Plunge the telescope 
and revolve the instrument about its vertical axis and again sight 
at the upper mark. SA\ing down to the lower mark. If the 
wire again bisects it, the adjustment is perfect. If not, fix a 
point half-way between the two positions of the lower mark. 
The plane of this point, the upper point, and the center of the 
instrument is truly vertical. Adjust the axis to these upper and 
lower points as when using the plumb-line. 

3. To make the line of collimation perpendicular to the revolving 
axis of the telescope. With the instrument level and the telescope 
nearly horizontal point at some well-defined point at a distance 
of 200 feet or more. Plunge the telescope and establish a point 
in the opposite direction. Turn the whole instrument about the 
vertical axis until it again points at the first mark. Again 
plunge to '^direct position '' {i.e,, with the level-tube under 
the telescope). If the vertical cross- wire again points at the 
second mark, the adjustment is perfect. If not, the error is 
one-fourth of the distance between the two positions of the 
second mark. Loosen the capstan screw on one side of the 
telescope and tighten it on the other side until the vertical 
wire is set at the one-fourth mark. Turn the whole instrument 
by means of the tangent screw until the vertical wire is midway 
between the two positions of the second mark. Plunge the 
telescope. If the adjusting has been skillfully done, the cross- 
wire should come exactly to the first mark. As an '^ erecting 
eyepiece '' reinverts an image already inverted, the ring carr3^ing 
the cross-wires must be moved in the sa?ne direction as the 
apparent error in order to correct that error. 



546 RAILROAD CONSTRUCTION. 

The necessity for the third adjustment lies principally in the practice 
of producing a Une by plunging the telescope, but when this is required to 
be done with great accuracy it is always better to obtain the forward point 
by reversion (as described above for making the test) and take the meaii 
of the two forward points. Horizontal and vertical angles are practically 
unaffected by small errors of this adjustment, imless, in the case of hori- 
zontal angles, the vertical angles to the points observed are very different. 

Unnecessary motion of the adjusting-screws may sometimes be avoided 
by carefully establishing the forward point on line by repeated reversions 
of the instrument, and thus determining by repeated trials the exact amount 
of the error. Differences in the amount of error determined would be 
evidence of inaccuracy in manipulating the instrument, and would show 
that an adjustment based on the first trial would probably prove unsatis- 
factory. 

The 2d and 3d adjustments are mutually dependent. If either adjust- 
ment is badly out, the other adjustment cannot be made except as follows: 

(a) The second adjustment can be made regardless of the third when 
the lines to the high point and the low point make equal angles with the 
horizontal. 

(b) The third adjustment can be made regardless of the second when 
the front and rear points are on a level with the instrument. 

When both of these requirements are nearly fulfilled, and especially 
when the error of either adjustment is small, no trouble will be found in 
perfecting either adjustment on account of a small error in the other ad- 
justment. 

If the test for the second adjustment is made by means of the plumb- 
line and the vertical cross-wire intersects the line at all points as the tele- 
scope is raised or lowered, it not only demonstrates at once the accuracy 
of that adjustment, but also shows that the third adjustment is either 
perfect or has so small an error that it does not affect the second, 

4. To Mve the bubble of the telescope-level in the center of the 
tube when the line of collimation is horizontal. The line of colli- 
mation should coincide vAih. the optical axis of the telescope. 
If the object-glass and eyepiece have been properly centered, 
the previous adjustment will have brought the vertical cross- 
wire to the center of the field of view. The horizontal cross- 
\rlre should also be brought to the center of the field of view, 
and the bubble should be adjusted to it. 

a. Peg method. Set up the transit at one end of a nearly 
level stretch of about 300 feet. Clamp the telescope with its 
bubble in the center. Drive a stake vertically under the eye- 
piece of the transit, and another about 300 feet away. Observe 
the height of the center of the eyepiece (the telescope being 
level) above the stake (calling it a); observe the reading of the 
rod when held on the other stake (calling it b) ; take the instru- 
ment to the other stake and set it up so that the eyepiece is 



APPENDIX. 547 

vertically over the stake, observing the height, c ; take a reading 
on the first stake, calling it d. If this adjustment is perfect, 
then 

a—d^h — c, 
or (a-c?)-(6-c)=0. 
Call (a-d)-(6-c)-2m. 
When m is positive, the line points downward; 
" m '* negative, " '' '' upward. 

To adjust: if the line points wp^ sight the horizontal cross- 
wire (by moving the vertical tangent screw) at a point which is 
m lower, then adjust the bubble so that it is in the center. 

By taking several independent values for a, 6, c, and c?, a mean value 
for m is obtained, which is more reliable and which may save much un- 
necessary working of the adjusting-screws. 

h. Using an auxiliary level. When a carefully adjusted level 
is at hand, this adjustment may sometimes be more easily 
made by setting up the transit and level, so that their lines of 
collimation are as nearly as possible at the same height. If a 
point may be found which is half a mile or more away and 
which is on the horizontal cross-wire of the level, the horizontal 
cross-wire of the transit may be pointed directly at it, and the 
bubble adjusted accordingly. Any slight difference in the 
heights of the lines of collimation of the transit and level (say 
yO may almost be disregarded at a distance of J mile or more, 
or, if the difference of level would have an appreciable effect, 
even this may be practically eliminated by making an estimated 
allowance when sighting at the distant point. Or, if a distant 
point is not available, a level-rod with target may be used at a 
distance of (say) 300 feet, making allowance for the carefully 
determined difference of elevation of the two lines of collimation. 

5. Zero of vertical circle. When the line of collimation is truly 
horizontal and the vertical axis is truly vertical, the reading 
of the vertical circle should be 0°. If the arc is adjustable, 
it should be brought to 0°. If it is not adjustable, the index 
error should be observed, so that it may be applied to all readings 
of vertical angles. 

ADJUSTMENTS OF THE WYE LEVEL. 

1. To make the line of collimation coincide with the center of 
the rings. Point the intersection of the cross-wires at some 



548 RAILROAD CONSTRUCTION. 

v^ell-defined point which is at a considerable distance. The in- 
strument need not be level, which allows much greater liberty 
in choosing a convenient point. The vertical axis should be 
clamped, and the clips over the wyes should be loosened and 
raised. Rotate the telescope in the wyes. The intersection of 
the cross-wires should be continually on the point. If it is not^ 
it requires adjustment. Rotate the telescope 180° and adjust 
one-half of the error b}^ means of the capstan-headed screws that 
move the cross-wire ring. It should be remembered that, with 
an erecting telescope, on account of the inversion of the image, 
the ring should be moved in the direction of the apparent error. 
Adjust the other half of the error with the leveling-screws. 
Then rotate the telescope 90° from its usual position, sight 
accurately at the point, and then rotate 180° from that position 
and adjust any error as before. It may require several trials, 
but it is necessary to adjust the ring until the intersection of 
the cross-wires will remain on the point for any position of 
rotation. 

If such a test is made on a very distant point and again on a point only 
10 or 15 feet from the instrument, the adjustment may be found correct 
for one point and incorrect for the other. This indicates that the object- 
slide is improperly centered. Usually this defect can only be corrected by 
an instrument-maker. If the difference is very small it may be ignored, 
but the adjustment should then be made on a point which is at about the 
mean distance for usual practice — say 1.50 feet. 

If the whole image appears to shift as the telescope is rotated, it indi- 
cates that the eyepiece is improperly adjusted. This defect is likewise 
usually corrected only by the maker. It does not interfere with instru- 
mental accuracy, but it usually causes the intersection of the cross-wires 
to be eccentric with the field of view. 

2. To make the axis of the level-tube parallel to the line of colli- 
mation. Raise the clips as far as possible. Swing the level 
so that it is parallel to a pair of opposite leveling-screws and 
clamp it. Bring the bubble to the middle of the tube by means 
of the leveling-screws. Take the telescope out of the wyes and 
replace it end for end, using extreme care that the wyes are not 
jarred by the action. If the bubble does not come to the center, 
correct one-half of the error by the vertical adjusting-screws at 
one end of the bubble. Correct the other half by the leveling- 
screws. Test the work by again changing the telescope end for 
end in the wyes. 

Care should be taken while making this adjustment to see 



APPENDIX. 549 

that the level-tube is vertically under the telescope. With the 
bubble in the center of the tube, rotate the telescope in the wyes 
for a considerable angle each side of the vertical. If the first 
half of the adjustment has been made and the bubble moves, it 
shows that the axis of the W3^es and the axis of the level-tube 
are not in the same vertical plane although both have been made 
horizontal. By moving one end of the level-tube sidewise by 
means of the horizontal screws at one end of the tube, the two 
axes may be brought into the same plane. As this adjustment 
is liable to disturb the other, both should be alternately tested 
until both requirements are complied with. 

By these methods the axis of the bubble is made parallel to 
the axis of the wyes; and as this has been made parallel to the 
lines of colhmation by means of the previous adjustment, the 
axis of the bubble is therefore parallel to the line of collimation. 

3. To make the line of collimation perpendicular to the vertical 
axis. Level up so that the instrument is approximately level 
over both sets of leveling-screws. Then, after leveling carefully 
over one pair of screws, revolve the telescope 180° If it is not 
level, adjust half of the error by means of the capstan-headed 
screw under one of the wyes, and the other half by the leveling- 
screws. Reverse again as a test. 

When the first two adjustments have been accurately made, good level- 
ing may always be done by bringing the bubble to the center by means of 
the leveling-screws, at every sight if necessary, even if the third adjust- 
ment is not made . Of course this third adjustment should be made as a 
matter of convenience, so that the line of collimation may be always level 
no matter in what direction it may be pointed, but it is not necessary to 
stop work to make this adjustment every time it is found to be defective. 

ADJUSTMENTS OF THE DUMPY LEVEL. 

1. To make the axis of the level-tube perpendicular to the vertical 
axis. Level up so that the instrument is approximately level 
over both sets of leveling-screws. Then, after leveling care- 
fully over one pair of screws, revolve the telescope 180°. If 
it is not level, adjust one-half of the error by means of the adjust- 
ing-screws at one end of the bubble, and the other half by 
means of the leveling-screws. Reverse again as a test. 

2. To make the line of collimation perpendicular to the vertical 
axis. The method of adjustment is identical with that for 
the transit (No. 4, p. 505) except that the cross- wire must be 



550 RAILROAD CONSTRUCTION. 

adjusted to agree with the level-bubble rather than vice versa, as 
is the case with the corresponding adjustment of the transit ; 
i.e., with the level-bubble in the center, raise or lower the hori- 
zontal cross-wire until it points at the mark known to be on 
a level with the center of the instrument. 

If the instrument has been well made and has not been dis- 
torted by rough usage, the cross-wires will intersect at the 
center of the field of view when adjusted as described. If they 
do not, it indicates an error which ordinarily can only be cor- 
rected by an instrument-maker. The error may be due to any 
one of several causes, which are 

(a) faulty centering of object-slide; 

(b) faulty centering of eyepiece; 

(c) distortion of instrument so that the geometric axis of 
the telescope is not perpendicular to the vertical axis. If the 
error is only just perceptible, it will not probably cause any 
error in the work. 



EXPLANATORY NOTE ON THE USE OF THE TABLES. 

The logarithms here given are "five-place/' but the last 
figure sometimes has a special mark over it {e.g., 6) which Indi- 
cates that one-half a unit in the last place should be added. 
For example 



the value 
.69586 
.69586 



includes all values between 
.6958575000 + and .6958624999. 
.6958625000 + and .6958674999. 



The maximum error in any one value therefore does not 
exceed one-quarter of a fifth-place unit. 

When adding or subtracting such logarithms allow a half-unit 
for such a sign. For example 

.69586 .69586 .6958^ 

.10841 .10841 .10841 

.12947 .1294? .12947 

.93375 .93375 .93375 

All other logarithmic operations are performed as usual and 
are supposed to be understood by the student. 

551 











TABLE I.— RADII OF CURVES. 








Deg 


0° 


1° 


2° 


3° 


Deg 


Min 


Radius. 


LogiJ 


Radius. 


iogH 


Radius. 


logH 


Radius. 


logJR 


Min 





00 


00 


5729-6 


3- 75813 


2864. 9 


3-45711 


1910-1 


3-28105 





1 


343775 


5.53627 


56357 


-75095 


2841.3 


-45351 


1899-5 


-27864 


1 


2 


171887 


5.23524 


5544-8 


74389 


2818-0 


-44993 


1889-1 


•27625 


2 


3 


114592 


5.05915 


5456-8 


.73694 


2795-1 


-44639 


1878-8 


.27387 


3 


4 


85944 


4-93421 


5371-6 


.73010 


2772-5 


-44287 


1868-6 


•27151 


4 


5 


68755 


4-83730 


5288-9 


-72336 


2750-4 


-43939 


1858-5 


•26915 


5 


6 


57296 


4-75812 


5208-8 


3-71673 


2728-5 


3-43593 


1848-5 


3^26681 


6 


7 


49111 


-69117 


5131-0 


-71020 


2707^0 


-43249 


1838-6 


-26448 


7 


8 


42972 


-63318 


5055-6 


-70377 


2685^9 


•42909 


18288 


•26217 


8 


9 


38197 


-58203 


4982-3 


-69743 


2665^1 


•42571 


1819-1 


.25986 


9 


10 


34377 


-53627 


4911-2 


-69118 


2644-6 


-42235 


1809-6 


-25757 


10 


11 


31252 


4-49488 


4842-0 


3-68502 


2624-4 


3-41903 


1800-1 


3-25529 


11 


12 


28648 


-45709 


4774-7 


-67895 


2604-5 


-41572 


1790-7 


•25303 


12 


13 


26444 


-42233 


4709-3 


•67296 


2584-9 


-41245 


1781-5 


•25077 


13 


14 


24555 


-39014 


4645-7 


-66705 


2565-6 


-40919 


1772-3 


•24853 


14 


15 


22918 


-36018 


4583- 8 


-66122 


2546- 6 


-40597 


1763-2 
1754-2 


-24629 


15 


16 


21486 


4-33215 


4523-4 


3-65547 


2527-9 


3-40276 


3-24407 


16 


17 


20222 


.30582 


4464-7 


-64979 


2509-5 


•39958 


1745-3 


-24186 


17 


18 


19099 


.28100 


4407-5 


-64419 


2491-3 


-39642 


1736-5 


-23967 


18 


19 


18093 


.25752 


4351-7 


-63865 


2473-4 


-39329 


1727-8 


-23748 


19 


20 


17189 


-23524 


4297-3 


-63319 


2455-7 


-39017 


1719-1 


-23530 


20 


21 


16370 


4-21405 


4244-2 


3-62780 


2438-3 


3-38708 


1710-6 


3-23314 


21 


22 


15626 


-19385 


4192.5 


■62247 


2421-1 


-38401 


1702-1 


-23098 


22 


23 


14947 


-17454 


4142-0 


-61720 


2404-2 


-38097 


1693-7 


-22884 


23 


24 


14324 


-15606 


4092-7 


-61200 


2387-5 


-37794 


1685.4 


-22670 


24 


25 


13751 


13833 
4-12130 


4044-5 


-60686 


2371-0 


37494 


1677-2 


-22458 


25 


26 


13222 


3997-5 


3-60178 


2354-8 


3-37195 


1669-1 


3.22247 


26 


27 


12732 


.10491 


3951-5 


-59676 


2338. 8 


.36899 


1661-0 


.22037 


27 


28 


12278 


.08911 


3906-6 


-59180 


23230 


-36604 


1653-0 


•21827 


28 


29 


11854 


.07387 


3862-7 


-58689 


2307-4 


-36312 


1645-1 


•21619 


29 


30 


11459 


-05915 
4-04491 


3819-8 


-58204 


2292-0 


-36021 


1637-3 


.21412 


30 


31 


11090 


3777-9 


3-57724 


2276-8 


3- 35733 


1629-5 


3.21206 


31 


32 


10743 


-03112 


37368 


-57250 


2261-9 


•35446 


1621-8 


.21000 


32 


33 


10417 


•01776 


3696-6 


-56780 


2247-1 


.35162 


1614-2 


.20796 


33 


34 


10111 


4 00479 


3657-3 


-56316 


2232-5 


-34879 


1606-7 


-20593 


34 


35 


9822-2 


399221 


3618-8 


-55856 


2218-1 


-34598 


1599-2 


.20390 


35 


36 


9549 


3 


3. 97997 


3581-1 


3-55401 


2203-9 


3-34318 


1591-8 


3-20189 


36 


37 


9291 


3 


-96807 


3544-2 


-54951 


21898 


-34041 


1584-5 


.19988 


37 


38 


9046 


•7 


-95649 


3508-0 


-54506 


2176-0 


-33765 


1577-2 


-19789 


38 


39 


8814 


8 


-9452 . 


3472-6 


-54065 


2162-3 


-33491 


1570-0 


•19590 


39 


40 


8594 


4 


-93421 


3437-9 


-53629 


2148-8 
2135-4 


-33219 
3-32949 


1562-9 


•19392 


40 


41 


8384 


8 


3-92349 


3403-8 


3-53197 


1555-8 


3-19195 


41 


42 


8185 


2 


-91302 


3370-5 


-52769 


2122-3 


-32680 


15488 


•18999 


42 


43 


7994 


8 


-90281 


3337-7 


-52345 


2109-2 


-32412 


1541-9 


•18804 


43 


44 


7813 


1 


-89282 


33057 


-51925 


2096-4 


-32147 


1535-0 


•18610 


44 


45 


7639 


5 


-88306 


3274-2 


-51510 


2083-7 


•31883 


1528-2 


-18417 


45 


46 


7473 


4 


3-87352 


3243-3 


3-51098 


2071-1 


3^31621 


1521-4 


3-18224 


46 


47 


7314 


4 


86418 


3213-0 


-50691 


2058-7 


•31360 


1514-7 


-18032 


47 


48 


7162 





-85503 


3183-2 


-50287 


2046-5 


•31101 


1508-1 


-17842 


48 


49 


7015 


9 


84608 


3154-0 


-49883 


2034-4 


•30843 


1501-5 


•17652 


49 


50 


6875 


6 


-83731 


3125-4 


-49490 


2022-4 


•30587 


1495-0 


-17462 
3-17274 


50 


51 


6740 


7 


3-82871 


3097-2 


3-49097 


2010-6 


3-30332 


1488-5 


51 


52 


6611 


1 


.82027 


3069-6 


-48707 


1998-9 


•30079 


1482-1 


•17087 


52 


53 


6486 


4 


.81200 


3042-4 


.48321 


1987-3 


•29827 


1475-7 


•16900 


53 


54 


6366 


3 


.80388 


3015-7 


.47939 


1975-9 


•29577 


1469-4 


•16714 


54 


55 


6250 


5 


.79591 


2989-5 


.47559 


1964-6 


-29328 


1463-2 


-16529 


55 


56 


6138 


9 


3-78809 


2963-7 


3.47183 


1953-5 


3-29081 


1457-0 


3-16344 


56: 


57 


6031. 


2 


-78040 


2938-4 


•46811 


1942.4 


•28835 


1450-8 


•16161 


57 


58 


5927. 


2 


•77285 


2913-5 


•46441 


1931.5 


•28590 


1444-7 


•15978 


58 


59 


5826. 


8 


•76542 


2889-0 


•46075 


1920-7 


•28347 


1438-7 


-15796 


59 


60 


5729.6 


•75813 


2864-9 


.45711 


1910-1 


-28105 


1432-7 


•15615 


60 



552 













TABLE 


I. 


-RADII OF CURVES. 












Deg 


4° 


5° 


6° 


7° 


Deg 


Min 


Radius. 


logM 


Radius. 


log It 


Radius. 


LogiJ 


Radius. 


LogJB 


Min 



1 
2 
3 
4 
5 


1432 
1426 
1420 
1415 
1409 
1403 


7 
7 
8 

2 
5 


3 


15615 
15434 
15255 
15076 
14897 
14720 


1146 
1142 
1138 
1134 
1131 
1127 


3 

5 
7 
9 
2 
5 


3 


05929 
05784 
05640 
05497 
05354 
05211 


955 
952 
950 
947 
944 
942 


37 
72 
09 
48 
88 
29 


2 


98017 
97896 
97776 
97657 
97537 
97418 


819 
817 
815 
813 
811 
809 


02 
08 
14 
22 
30 
40 


2. 


91329 
91226 
91123 
91021 
90918 
90816 




1 
2 
3 
4 
5 


6 
7 
8 
9 
10 


1397 
1392 
1386 
1380 
1375 


8 

1 
5 
9 
4 


3 


14543 
14367 
14191 
14017 
13843 


1123 
1120 
1116 
1112 
1109 


8 

2 
5 
9 
3 


3 


05069 
04928 
04787 
04646 
04506 


939 
937 
934 
932 
929 


72 
16 
62 
09 
57 


2 


97300 
97181 
97063 
96945 
96828 


807 
805 
803 
801 
800 


50 
61 
73 
86 
00 


2. 

* 


90714 
90612 
90511 
90410 
90309 


6 
7 
8 
9 
10 


11 
12 
13 
14 
15 


1369 
1364 
1359 
1353 
1348 


9 
5 
1 
8 
4 


3 


13669 
13497 
13325 
13154 
12983 


1105 
1102 
1098 
1095 
1091 


8 

2 
7 
2 
7 


3 


04366 
04227 
04088 
03949 
03811 


927 
924 
922 
919 
917 


07 
58 
10 
64 
19 


2 


96711 
96594 
96478 
96361 
96246 


798 
796 
794 
792 
790 


14 
30 
46 
63 
81 


2 


90208 
90107 
90007 
89907 
89807 


11 
12 
13 
14 
15 


16 
17 
18 
19 
20 


1343 
1338 
1332 
1327 
1322 


2 

8 
6 

5 


3 


12813 
12644 
12475 
12307 
12140 


1088 
1084 
1081 
1078 
1074 


3 
8 

4 
1 
7 


3 


03674 
03537 
03400 
03264 
03128 


914 
912 
909 
907 
905 


75 
33 
92 
52 
13 


2 


96130 
96015 
95900 
95785 
95671 


789 
787 
785 
783 
781 


00 
20 
41 
62 
84 


2 


89708 
89608 
89509 
89410 
89312 


16 
17 
18 
19 
20 


21 
22 
23 
24 
25 


1317 
1312 
1307 
1302 
1297 


5 
4 
4 
5 
6 


3 


11974 
11808 
11642 
11477 
11313 


1071 
1068 
1064 
1061 
1058 


3 

7 
4 
2 


3 


02992 
02857 
02723 
02589 
02455 


902 
900 
898 
895 
893 


76 
40 
05 
71 
39 


2 


95557 
95443 
95330 
95217 
95104 


780 
778 
776 
774 
773 


07 
31 
55 
81 
07 


2 


89213 
89115 
89017 
88919 
88821 


21 

22 
23 
24 
25 


26 
27 
28 
29 
30 


1292 
1287 
1283 
1278 
1273 


7 
9 
1 
3 
6 


3 


11150 
10987 
10825 
10663 
10502 


1054 
1051 
1048 
1045 
1042 


9 
7 
5 
3 

1 


3 


02322 
02189 
02056 
01924 
01792 


891 
888 
886 
884 
881 


08 
78 

49 
21 
95 


2 


94991 
94879 
94767 
94655 
94544 


771 

769 

767 

766. 

764. 


34 
61 
90 
19 
49 


2 


88724 
88627 
88530 
88433 
88337 


26 
27 
28 
29 
30 


31 
32 
33 
34 
35 


1268 
1264 
1259 
1255 
1250 


9 
2 
6 

4 


3 


10341 
10182 
10022 
09864 
09705 


1039 
1035 
1032 
1029 
1026 



9 
8 
7 
6 


3 


01661 
01530 
01400 
01270 
01140 


879 
877 
875 
873 
870 


69 
45 
22 
00 
80 


2 


94433 
94322 
94212 
94101 
93991 


762. 
761. 
759. 
757. 
756. 


80 
11 
43 
76 
10 


2. 


88241 
88145 
88049 
87953 
87858 


31 
32 
33 
34 
35 


36 
37 
38 
39 
40 


1245 
1241 
1236 
1232 
1228 


9 

4 
9 
5 

1 


3 


09548 
09391 
09234 
09079 
08923 


1023 
1020 
1017 
1014 
1011 


5 
5 
5 
5 
5 


3 


01010 
00882 
00753 
00625 
00497 


868 
866 
864 
862 
859 


60 
41 
24 
07 
92 


2 


93882 
93772 
93663 
93554 
93446 


754. 
752. 
751. 
749. 
747. 


44 
80 
16 
52 
89 


2. 


87762 
87668 
87573 
87478 
87384 


36 
37 
38 
39 
40 


41 
42 
43 
44 
45 


1223 
1219 
1215 
1210 
1206 


7 
4 
1 
8 
6 


3 


08769 
08614 
08461 
08308 
08155 


1008 

1005 

1002 

999 

996 


6 

6 

7 

76 

87 


3 

3 

2 


00370 
00242 
00116 
99989 
99863 


857 
855 
853 
851 
849 


78 
65 
53 

42 
32 


2 


93337 
93229 
93122 
93014 
92907 


746. 
744. 
743. 
741. 
739. 


27 
66 
06 
46 
86 


2. 


87290 
87196 
87102 
87008 
86915 


41 

42 
43 
44 
45 


46 
47 
48 
49 
50 


1202 
1198 
1194 
1189 
1185 


4 
2 

9 
8 


3 


08003 
07852 
07701 
07550 
07400 


993 
991 
988 
985 
982 


99 

13 
28 
45 
64 


2 


99738 
99613 
99488 
99363 
99239 


847 
845 
843 
841 
838 


23 
15 
08 
02 
97 


2 


92800 
92693 
92587 
92480 
92374 


738 
736 
735 
733 
732 


28 
70 
13 
56 
01 


2 


86822 
86729 
86636 
86544 
86451 


46 
47 
48 
49 
50 


51 
52 
53 
54 
55 


1181 
1177 
1173 
1169 
1165 


7 
7 
6 
7 
7 


3 


07251 
07102 
06954 
06806 
06658 


979 
977 
974 
971 
968 


84 
06 
29 
54 
81 


2 


99115 
98992 
98869 
98746 
98624 


836 
834 
832 
830 
828 


93 
90 
89 
88 
88 


2 


92269 
92163 
92058 
91953 
91849 


730 
728 
727 
725 
724 


45 
91 
37 
84 
31 


2 


86359 
86267 
86175 
86084 
85992 


51 
52 
53 
54 
55 


56 
57 
58 
59 
60 


1161 
1157 
1154 
1150 
1146 


8 
9 

1 
3 


3 


06511 
06365 
06219 
06074 
05929 


966 
963 
960 
958 
955 


09 
39 
70 
03 
37 


2 


98501 
98380 
98258 
98137 
98017 


826 
824 
822 
820 
819 


89 
91 
93 
97 
02 


2 

1 


91744 
91640 
91536 
91433 
91329 


722 
721 
719 
718 
716 


79 
28 
77 
27 
.78 


2 


85901 
•85810 
.85719 
•85629 
.85538 


56 
57 
58 
59 
60 



553 













TABLE 


I. 


—RADII OF CURVES. 












Deg. 


8° 


9^ 


10° 


11° 


Deg 


Min. 


Radius. 


logJt 


Radius. 


logB 


Radius. 


Log-B 


Radius. 


logn 


Min 




1 
2 
3 
4 
5 


716 
715 
713 
712 
710 
709 


78 

.29 
81 
34 
87 
40 


2 


85538 
.85448 
85358 
85268 
85178 
85089 
85000 
84911 
84822 
84733 
84644 


637 
636 
634 
633 
632 
631 


27 
10 
93 
76 
60 
44 


2 


80432 
80352 
86272 
80192 
80113 
80033 


573 
572 
571 
570 
569 
568 


69 
73 
78 
84 
90 
96 


2 


75867 
75795 
75723 
75651 
75579 
75508 


521 
520 
520 
519 
518 
517 


67 
88 

10 
32 
54 
76 


2 


71739 
71674 
71608 
71543 
71478 
71413 




1 
2 
3 
4 
5 


6 
7 
8 
9 
10 


707 
706 
705 
703 
702 


95 
49 
05 
61 
17 


630 
629 
627 
626 
625 


29 
14 
99 
85 
71 


2 


79954 
79874 
79795 
79716 
79637 


568 
567 
566 
565 
564 


02 
09 
16 
23 
31 


2 


75436 
75365 
75293 
75222 
75151 


516 
516 
515 
514 
513 


99 
21 
44 
68 
91 


2 


71348 
71283 
71218 
71153 
71088 


6 
7 
8 
9 
10 


11 
12 
13 
14 
15 


700 
699 
697 
696 
695 


75 
33 
91 
50 
09 


2 


84556 
84468 
84380 
84292 
84204 


624 
623 
622 
621 
620 


58 

45 
32 
20 
09 


2 


79558 
79480 
79401 
79323 
79245 


563 
562 
561 
560 
559 


38 

47 
55 
64 
73 


2 


75080 
75009 
74939 
74868 
74798 


513 
512 
511 
510 
510 


15 
38 
63 
87 
11 


2 


71024 
70959 
70895 
70831 
70767 


11 
12 
13 
14 
15 


16 
17 
18 
19 
20 


693 
692 
690 
689 
688 


70 
30 
91 
53 
16 


2 


84117 
84029 
83942 
83855 
83768 


618 
617 
616 
615 
614 


97 
87 
76 
66 
56 


2 


79167 
79089 
79011 
78934 
78856 


558 
557 
557 
556 
555 


82 
92 
02 
12 
23 


2 


74727 
74657 
74587 
74517 
74447 


509 
508 
507 
507 
506 


36 
61 
86 
12 
38 


2 


70702 
70638 
70575 
70511 
70447 


18 
17 
18 
19 
20 


21 
22 
23 
24 
25 


686- 
685. 
684. 
682. 
681. 


78 
42 
06 
70 
35 


2 
2~ 


83682 
83595 
83509 
83423 
83337 
83251 
83166 
83080 
82995 
82910 


613 
612 
611 
610 
609 

608 
606 
605 
604 
603 


47 
38 
30 
21 
14 
06 
99 
93 
86 
80 


2 


78779 
78702 
78625 
78548 
7847i 


554 
553 
552 
551 
550 


34 
45 
56 
68 
80 


2 


74377 
74307 
74238 
74168 
74099 


505 
504 
504 
503 
502 


64 
90 
16 
42 
69 


2 


70383 
70320 
70257 
70193 
70130 


21 
22 
23 
24 
25 


26 
27 
28 
29 
30 


680. 
678. 
677. 
676- 
674. 


01 
67 
34 
01 
69 


2 


78395 
78318 
78242 
78165 
78089 


549 
549 
548 
547 
546 


92 
05 
17 
30 
44 


2 


74030 
73961 
73892 
73823 
73754 


501 
501 
500 
499 
499 


96 
23 
51 
78 
06 


2 


70067 
70004 
69941 
69878 
69815 


26 
27 
28 
29 
30 


31 
32 
33 
34 
35 


673. 
672. 
670- 
669. 
668. 


37 
06 
75 
45 
15 


2 


82825 
82740 
82656 
82571 
82487 


602 
601 
600 
599 
598 


75 
70 
65 
61 
57 


2 
T 


78013 
77938 
77862 
77786 
77711 

77636 
77561 
77486 
77411 
77336 


545 
544 
543 
543 
542 


57 
71 
86 
00 
15 


2 


73685 
73617 
73548 
73480 
73412 


498 
497 
496 
496 
495 


34 
62 
91 
19 
48 


2 


69752 
69690 
69627 
69565 
69503 


31 
32 
33 
34 
35 


36 
37 
38 
39 
40 


666. 
665. 
664. 
663. 
661. 


86 
57 
29 
01 
74 


2 


82403 
82319 
82235 
82152 
82068 


597 
596 
595 
594 
593 
592 
591 
590 
589 
588 


53 
50 
47 
44 
42 

40 
38 
37 
36 
36 


541 
540 
539 
538 
537 


30 
45 
61 
76 
92 


2 


73343 
73275 
73207 
.73140 
73072 


494 
494 
493 

492 
491 


77 
07 
36 
66 
96 


2 


69440 
69378 
69316 
69254 
69192 


36 
37 
38 
39 
40 


41 
42 
43 
44 
45 


660. 
659. 
657- 
656. 
655. 


47 
21 
95 
69 
45 


2 


81985 
81902 
81819 
81736 
81653 


2 


77261 
77187 
77112 
77038 
76964 


537 
536 
535 
534 
533 


09 
25 
42 
59 
77 


2 


73004 
72937 
72869 
72802 
72735 


491 
490 
489 
489 
488 


26 
56 
86 
17 
48 


2 


69131 
69069 
69007 
68946 
68884 


41 
42 
43 
44 
45 


46 
47 
48 
49 
50 


654. 
652. 
651- 
650. 
649. 


20 
96 
73 
50 
27 


2 


81571 
81489 
81406 
81324 
81243 


587 
586 
585 
584 
583 


36 
36 
36 
37 
38 


2 


76890 
76816 
76742 
76669 
76595 


532 
532 
531 
530 
529 


94 
12 
30 
49 
67 


2 


72668 
72601 
72534 
72467 
72401 


487 
487 
486 
485 
485 


79 
10 
42 
73 
05 


2 


68823 
68762 
68701 
68640 
68579 


46 
47 
48 
49 
50 


51 
52 
53 
54 
55 


648. 
646. 
645. 
644. 
643- 


05 
84 
63 
42 
22 


2 


81161 
81079 
80998 
80917 
80836 


582 
581 
580 
579 
578 


40 
42 
44 
47 
49 


2 


76522 
76449 
76376 
76303 
76230 


528 
528 
527 
526 
525 


86 
05 
25 
44 
64 


2 


72334 
72267 
72201 
72135 
72069 


484 
483 
483 
482 
481 


37 
69 
02 
34 
67 


2 


68518 
68457 
68396 
68335 
68275 


51 
52 
53 
54 
55 


56 
57 
58 
59 
dO 


642. 

640 

639 

638 

637 


02 
83 
64 
45 
27 


2 


80755 
80674 
80593 
80513 
80432 


577 
576 
575 
574 
573 


53 
56 
60 
64 
69 


2 


76157 
76084 
76012 
75939 
75867 


524 
524 
523 
522 
S21 


84 
05 
25 
46 
67 


2 


72003 
71937 
71871 
71805 
71739 


481 
480 
479 
479 
478 


00 
33 
67 
00 
34 


2 


68214 
68154 
68094 
68035 
6797S 


56 
57 
58 
59 
60 



554 













TABLE I. 


-RADII OF CURVES. 










Deg. 


Radius. 


Log^ 


Deg. 


Radius. 


LogiJ 


Deg. 


Radius. 


Log 12 


Deg. 


Radius 


logM 


12° 

2 
4 
6 
8 
10 
12 
14 
16 
18 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42 
44 
46 

50 
52 
54 
56 
§8 

2 
4 
6 
8 
10 
12 
14 
16 
18 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 


478 
477 
475 
474 
473 


34 
02 
71 
40 
10 


2 


67973 
67853 
67734 
67614 
67495 


14° 

2 
4 
6 
8 
10 
12 
14 
16 
18 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42 
44 
46 
48 


410 
409 
408 
407 
406 


.28 
31 
34 
38 
42 


2 


61307 
61205 
61102 
61000 
60898 


16° 

5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

17° 

5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
55 

18° 
5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

19° 

5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

30° 

5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 
21° 


359 
357 
355 
353 
351 
350 


26 
42 
59 
77 
98 
21 


2 


55541 
55317 
55094 
54872 
54652 
54432 


21° 

10 
20 
30 
40 
50 


274 
272 
270 
268 
266 
264 


• 37 
23 
13 
06 
02 
02 


2-43833 
.43494 
•43157 
•42823 
•42492 


471 
470 
469 
467 
466 


81 
53 
25 
98 
72 


2 


67376 
67258 
67140 
67022 
66905 


405 
404 
403 
402 
401 


47 
53 
58 
65 
71 


2 


60796 
60694 
60593 
60492 
60391 


•42163 


348 
346 
344 
343 
341 
339 


45 
71 
99 
29 
60 
93 


2 


54214 
53997 
53780 
53565 
53351 
53138 


22° 
10 
20 
30 
40 
50 


262 
260 
258 
256 
254 
252 


04 
10 
18 
29 
43 
60 


2-41837 
•41513 
•41192 
■40873 


465 
464 
462 
461 
460 


46 
21 
97 
73 
50 


2 


66788 
66671 
66555 
66439 
66323 


400 
399 
398 
398 
397 


78 
86 

94 
02 
11 


2 


60291 
60190 
60090 
59990 
59891 


•40557 
•40243 


338 
336 
335 
333 
331 
330 

328 

327 
325 
324 
322 
321 


27 
64 
01 
41 
82 
24 
68 
13 
60 
09 
59 
10 


2 


52927 
52716 
52506 
52297 
52090 
51883 


23° 

10 
20 
30 
40 
50 

24° 
10 
20 
30 
40 
50 

25° 

30 
26° 

30 

27° 

30 

28° 

30 

29° 

30 
30° 

31° 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

52 

54 

56 

58 

60 


250 
249 
247 
245 

243 
242 


79 
01 
26 
53 
82 
14 


2-39931 
•39622 
•39315 


459 
458 
456 
455 
454 


28 
06 
85 

65 

45 


2 


66207 
66092 
65977 
65863 
65748 


396 
395 
394 
393 
392 

391 
390 
389 
389 
388 


20 
30 
40 
50 
61 
72 
84 
96 
08 
21 


2 


59791 
59692 
59593 
59494 
59398 


•39019 
•38707 
3840'? 


2 


51677 
51472 
51269 
51066 
50864 
50663 


240 
238 
237 
235 
234 
232 


49 
85 
24 
65 
08 
54 


2.38109 
■37813 


453 
452 
450 
449 
448 


26 
07 
89 
72 
56 


2 


65634 
65521 
65407 
65294 
65181 


2 


59298 
59199 
59102 
59004 
58907 


.37519 
.37227 
36937 
•36649 


319 
318 
316 
315 
313 
312 

311 
309 
308 
306 
305 
304 
302 
301 
300 
299 
297 
296 


62 
16 
71 
28 
86 
45 
06 
67 
30 
95 
60 
27 
94 
63 
33 
04 
77 
50 


2 


50464 
50265 
50067 
49869 
49673 
49478 


231 
226 
222 
218 


01 
55 
27 
15 


2 36363 


447 
446 
445 
443 
442 


40 
24 
09 
95 
81 


2 


65069 
64957 
64845 
64733 
64622 


50 
52 
54 
56 
58 

15° 

2 

4 

6 

8 

10 

12 

14 

16 

18 


387 
386 
385 
384 
383 


34 
48 

62 
77 
91 


2 


58809 
58713 
58616 
58519 
58423 


•35517 
•34688 
.33875 


214 
210 
206 
203 


18 
36 
68 
13 


2 •33078 
.32296 


441 
440 
439 
438 
437 
436 
435 
433 
432 
431 


68 
56 
44 
33 
22 
12 
02 
93 
84 
76 


2 


64511 
64400 
64290 
64180 
64070 


383 
382 
381 
380 
379 


06 
22 
38 
54 
71 


2 


58327 
58231 
58135 
58040 
57945 


2 


49284 
49090 
48898 
48706 
48515 
48325 


.31529 
•30776 


199 
196 
193 
190 


70 
38 
19 
09 


2.30037 
.2931(5 
.28597 


2 


63960 
63851 
63742 
63633 
63524 


378 
378 
377 
376 
375 


88 

05 
23 
41 
60 


2 


57850 
57755 
57661 
57566 
57472 


•27896 


2 


48136 
47948 
47760 
47573 
47388 
47203 


187 
181 
176 
171 
166 


10 
40 
05 
02 
28 


2.27207 
.25863 
.24563 
.23303 


430 
429 
428 
427 
426 


69 
62 
56 
50 
44 


2 


63416 
63308 
63201 
63093 
62986 


20 
22 
24 
26 
28 
30 
32 
34 
36 
38 


374 
373 
373 

372 
371 


79 
98 
17 
37 
57 


2 


5737§ 
57284 
57191 
57097 
57004 


.22083 


161 
157 
153 
149 
146 


80 
58 
58 
79 
19 


2.20899 
.19749 
.18633 
.17547 


295 
294 
292 
291 
290 
289 


25 
00 
77 
55 
33 
13 


2 


47018 
46835 
46652 
46471 
46289 
46109 


425 
424 
423 
422 
421 


40 
35 
32 
28 
26 


2 


62879 
62773 
62666 
62560 
62454 


370 
369 
369 
368 
367 
366 
366 
365 
364 
363 

363 
362 
361 
360 
360 


78 
99 
20 
42 
64 
86 
09 
31 
55 
78 
02 
26 
51 
76 
01 


2 


56911 
56819 
56726 
56634 
56542 


.16492 


142 
139 
136 
133 
130 


77 
52 
43 
47 
66 


2.15462 
.14464 
.13489 
.12539 


287 
286 
285 
284 
283 
282 


94 
76 
58 

42 
27 
12 


2 


45930 
45751 
45573 
45396 
45219 
45044 


40 
42 
44 
46 
48 
50 
52 
54 
56 
58 


420 
419 
418 
417 
416 
415 
414 
413 
412 
411 


23 
22 
20 
19 
_I9 
19 
20 
21 
23 
25 


2 


62349 
62243 
62138 
62034 
61929 


40 
42 
44 
46 
48 
50 
52 
54 
56 
58 

16° 


2 


56450 
56358 
56266 
56175 
56084 


.11613 


127 
125 
122 
120 
118 


97 
39 
93 
57 
31 


2.10709 
.09827 
.08965 
•08124 


280 
279 
278 
277 
276 
275 


99 
86 
75 
64 
54 
.45 


2 


44869 
44694 
44521 
44348 
44176 
44004 


2 


61825 
61721 
61617 
61514 
61410 


2 


55993 
55902 
55812 
55721 
55631 


07302 


114 
110 
106 
103 
100 


06 
13 
50 
13 
00 


2.05713 
.04192 
.02736 
.01340 


274 


.37 


2 


43833 


114° 


410 


28 


2 


61307 


359 


26 


2 


55541 


2.00000 



555 



TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS 
FOR A 1° CURVE. 



A 


T^g' 


Ext. 
Dist. 
21;. 


Lonff 

Chord 


A 


T^g. 


Ext. 

Dist. 


Long 
Chord 


A 

31° 

10 
20 

30 
40 
50 

10 
20 
30 
40 
50 


T^g. 


Ext. 
Dist. 


Lonff 
Chord 


1° 

10' 
20 
30 
40 
50 


50 
58 
66 
75 
83 
91 


.00 
.34 
.67 
.01 
.34 
• 68 










.218 
.297 
388 
.491 
.606 
.733 


100 
116 
133 
150 
166 
183 


.00 
.67 
.33 
.00 
.66 
33 


11° 

10 
20 
30 
40 
50 


551 
560 
568 
576 
585 
593 


• 70 
.11 
.53 
.95 
.36 
.79 


26 
27 
28 
28 
29 
30 


.500 
.313 
.137 
.974 
.824 
686 


1098 
1114 
1131 
1148 
1164 
1181 


.3 
.9 

.5 
.1 

■I 


1061. 9 
1070.6 
1079.2 
1087-8 
1096-4 
1105.1 


97 
99 
100 
102 
103 
105 


.58 

• 15 
.75 
.35 
.97 
.60 


2088 
2104 
2121 
2137 
2153 
2170 


3 
7 
1 

-4 
8 
2 


2° 
10 
20 
30 
40 
50 


100 
108 
116 
125 
133 
141 


• 01 
■ 35 
68 
02 
36 
70 





• 873 
.024 
188 
364 
552 
752 


199 
216 
233 
249 
266 
283 


99 
66 
32 
98 
65 
31 


13° 

10 
20 
30 
40 
50 
13° 
10 
20 
30 
40 
50 


602 
610 
619 
627 
635 
644 


• 21 
.64 
.07 
.50 
.93 
.37 


31 
32 
33 
34 
35 
36 


• 561 
447 
347 
259 
183 
120 


1197 
1214 
1231 
1247 
1264 
1280 


8 

4 

.5 
1 
7 


1113.7 
1122.4 
1131.0 
1139-7 
1148.4 
1157.0 


107 
108 
110 
112 
113 
115 


.2412186 
90 2202 
57 2219 
25 2235 
95 2251 
66 2268 


5 
9 
2 
6 
9 
3 


3° 

10 
20 
30 
40 
50 


150 
158 
166 
175 
183 
191 


04 
38 

72 
06 
40 
74 


1 

2 
2 
2 
2 
3 


964 
188 
425 
674 
934 
207 


299 
316 
333 
349 
366 
383 


97 
63 
29 
95 
61 
27 


652 
661 
669 
678 
686 
695 


.81 
.25 
.70 
.15 
.60 
06 


37 
38 
39 
39 
40 
42 


069|l297 
031il313 
006 1330 
993 1346 
992 1363 
004 1380 


2 
8 
3 
9 
4 



33° 

10 
20 
30 
40 
50 


1165.7 
1174.4 
1183.1 
1191.8 
1200.5 
1209.2 


117 
119 
120 
122 
124 
126 


38 2284 
12 2301 
87 2317 
63,2333 
41 2349 
2C:2366 


6 

3 
6 
9 
2 


4° 
10 
20 
30 
40 
50 


200 
208 
216 
225 
233 
241 


08 
43 
77 
12 
47 
81 


3 
3 
4 
4 
4 
5 


492 
790 
099 
421 
755 
100 


399 
416 
433 
449 
466 
483 


92 
58 
24 
89 
54 
20 


14' 

10 
20 
30 
40 
50 
15° 
10 
20 
30 
40 
50 

16° 

10 
20 
30 
40 
50 


703 
711 
720 
728 
737 
745 


.51 
97 
44 
90 
37 
85 


43 
44 
45 
46 
47 
48 


029 1396 
066 1413 
116 1429 
178 1446 
253 1462 
341 1479 


5 
1 
6 
2 
7 
2 


34° 

10 
20 
30 
40 
50 


1217.9 
1226-6 
1235.3 
1244.0 
1252.8 
1261.5 


128 
129 
131 
133 
135 
137 


OC 
82 
65 
50 
36 
23 


2382 
2398 
2415 
2431 
2447 
2464 


5 
8 
1 
4 
7 



5° 
10 
20 
30 
40 
50 


250 
258 
266 
275 
283 
291 


16 
51 
86 
21 
57 
92 


5 
5 
6 
6 
7 
7 


459 
829 
211 
606 
013 
432 


499 
516 
533 
549 
566 
583 


85 

50 
15 
80 
44 
09 


754 
762 
771 
779 
788 
796 


32 
80 
29 
77 
26 
75 


49 
50 
51 
52 
53 
55 


441 
554 
679 
818 
969 
132 


1495 
1512 
1528 
1545 
1561 
1578 


7 
3 
8 
3 
8 
3 


35° 

10 
20 
30 
40 
50 


1270.2 
1279.0 
1287.7 
1296.5 
1305.3 
1314.0 


139 
141 
142 
144 
146 
148 


11 
01 
93 
85 
79 
75 


2480 
2496 
2512 
2529 
2545 
2561 


2 
5 
8 

3 
5 


6° 

10 
20 
30 
40 
50 


300 
308 
316 
325 
333 
342 


28 
64 
99 
35 
71 
08 


7 
8 
8 
9 
9 
10 


863 
307 
762 
230 
710 
202 


599 
616 
633 
649 
666 
682 


73 
38 
02 
66 
30 
94 


805 
813 
822 
830 
839 
847 


25 56 
75 57 
25 58 
7659 
27,61 
78,62 


309 
498 
699 
914 
141 
381 


1594 
1611 
1627 
1644 
1660 
1677 


8 
3 
8 
3 
8 
3 


36° 

10 
20 
30 
40 
50 


1322-8 
1331.6 
1340-4 
1349.2 
13580 
136^8 


150 
152 
154 
156 
158 
160 


71 
69 
69 
70 
72 
76 


2577 
2594 
2610 
2626 
2642 
2658 


8 

3 
5 
7 
9 


7° 
10 
20 
30 
40 
50 


350. 
358. 
367. 
375. 
383- 
392. 


44' 
81 
17, 
54 

91; 

28 


10. 
11. 
11. 
12. 
12. 
13. 


707 
224 
753 
294 
847 
413 


699. 
716. 
732. 
749. 
766. 
782. 


57 
21 
84 
47 
10 
73 


17° 
10 
20 
30 
40 
50 


856 
864 
873 
881 
890 
898 


30 
82 
35 
88 
41 
95, 


63 
64 
66 
67 
68 
70 


634 
900 
178 
470 
774 
091 


1693 
1710 
1726 
1743 
1759 
1776 


8 
3 
8 

2 
7 
2 


37° 
10 
20 
30 
40 
50 


1375-6 
13844 
1393.2 
1402.0 
1410.9 
1419.7 


162 
164 
166 
169 
171 
173 


81 
87 
95 
04 
15 
27 


2675 
2691 
2707 
2723 
2739 
2756 


1 
3 
5 
7 
9 
1 


8° 
10 
20 
30 
40 
50 


400. 
409. 
417. 
425. 
434. 
442. 


66: 

03 
41 
79, 
17 
55 


13. 
14. 
15. 
15. 
16. 
17. 


991 
582 
184 
799 
426 
066 


799. 
815. 
832. 
849. 
865. 
882. 


36 
99 
61 
23 
85 
47 


18° 
10 
20 
30 
40 
50 

19° 

10 

20 

30 

40 

50 
20° 

10 

20 ! 

30 

40 

50 
31° 


907 
916. 
924. 
933. 
941. 
950. 


49' 
03 
58 
13 
69 
25 


71. 
72. 
74. 
75. 
76- 
78. 


421 
764 
119 
488 
869 
264 


1792 

1809. 

1825 

1842. 

1858. 

1874. 


6 
1 
5 

4 
9 


38° 
10 
20 
30 
40 
50 

39° 

10 
20 
30 
40 
50 

30° 

10 
20 
30 
40 
50 


1428.6 
1437.4 
1446.3 
1455.1 
1464.0 
1472.9 


175 
177 
179 
181 
184 
186 


41 
55 
72 
89 
08 
29 


2772 
2788 
2804 
2820 
2836 
2853 


3 
4 
6 
7 
9 



9° 

10 
20 
30 
40 
50 


450. 
459- 
467. 
476. 
484. 
492. 


93; 
32 
71 
10 
49 
88 


17. 
18. 
19. 
19. 
20. 
21. 


717 
381 
058 
746 
447 
161 


899. 
915. 
932. 
948. 
965. 
982. 


09 
70 
31 
92 
53 
14 


958 
967 
975 
984 
993 
1001 


81 
38 
96 
53 
12 
70 


79. 

81. 

82. 

83 

85 

86 


671 
092 
525 
972 
431 
904 


1891. 
1907. 
1924 
1940 
1957 
1973 


3 
8 

2 
6 
1 
5 


1481. 8 
1490.7 
1499.6 
1508.5 
1517.4 
1526.3 


188 
190 
192 
195 
197 
199 


51 2869 
74 2885 
99 2901 
25 2917 
5312933 
82 2949 


2 
3 
4 
81 
T 
Si 


10° 

10 
20 
30 
40 
50 


501. 
509 
518 
526 
534 
543 


28 
68 
08 
48 
89 
29 


21. 

22 
23 
24 
24 
25 


886 

624 
375 
138 
913 
700 


998. 
1015. 
1031 
1048 
1065 
1081 


74 
35 
95 
54 
14 
73 


1010 
1018 
1027 
1036 
1044 
1053 


29 
89 
49 
09 
70 
31 


88 
89 
91 
92 
94 
96 


389 
888 
399 

924 
462 
013 


1989 
2006 
2022 
2039 
2055 
2071 


9 
3 

7 
1 
5 
9 


1535.3 
1544.2 
1553.1 
1562.1 
1571.0 
1580.0 


202 
204 
206 
209 
211 
213 


12 2965 
44J2982 
77^2998 
12 3014 
48 3030 
86 3046. 


9' 
01 
1 
2 
2 
3 


li^ 


551 


70i26 


500 


1098 


33 


1061 


_93. 


97_ 


577 


2088. 


_3_ 


31° 


1589.0 


216. 


25 3062. 


i 



556 



TABLE II.- 



-TANGENTS, EXTERNAL DISTANCES, 
FOR A 1° CURVE. 



AND LONG CHORDS 



A 


Jang- 


Ext. 
Dist. 


Long 
Chord 


A 


T^g. 


Ext. 
Dist. 


Long 
Chord 
jLC. 


A 


T^ng. 


Ext. 
Dist. 


Long 
Chord 


31° 

10' 
20 
30 
40 
50 


1589 
1598 
1606 
1615 
1624 
1633 


.0 
.0 
9 
9 
.9 
9 


216 
218 
221 
223 
225 
228 


25 
66 
08 
51 
96 
42 


3062 
3078 
3094 
3110 
3126 
3142 


4 
4 
5 
5 
6 
6 


41° 

10 
20 
30 
40 
50 

43° 

10 
20 
30 
40 
50 

43° 

10 
20 
30 
40 
50 


2142.2387 
2151.7390 
2161.2394 
2170.8 397 
2180.3,400 
2189.9:404 


384013 
71 4028 
06 4044 
43 4059 
82 4075 
22 4091 


1 
7 
3 
9 
5 
1 


51° 

10 
20 
30 
40 
50 
53° 
10 
20 
30 
40 
50 


2732 
2743 
2753 
2763 
2773 
2784 


9 

1 
4 
7 
9 
2 


618 
622 
627 
631 
636 
640 


39 
81 
24 
69 
16 
66 


4933 

4948 
4963 
4978 
4993 
5008 


4 
4 
4 
4 
4 
4 


32° 

10 
20 
30 
40 
50 


1643 
1652 
1661 
1670 
1679 
1688 







1 
1 


230 
233 
235 
238 
240 
243 


90 
39 
90 
43 
96 
52 


3158 
3174 
3190 
3206 
3222 
3238 


6 
6 
6 
6 
6 
6 


2199.4407 
2209.0411 
2218.61414 
2228.l|417 
2237.71421 
2247-3 424 


64 4106 
07 4122 
524137 
99 4153 
48 4168 
98 4184 


6 
2 
7 
3 
8 
3 


2794 
2804 
2815 
2825 
2835 
2846 


5 
9 
2 
6 
9 
3 


645 
649 
654 
658 
663 
668 


17 
70 
25 
83 
42 
03 


5023 
5038 
5053 
5068 
5083 
5098 


4 
4 
4 
3 
3 
2 


33° 

10 
20 
30 
40 
50 


1697 
1706 
1715 
1724 
1733 
1742 


2 
3 
3 
4 
5 
6 


246 
248 
251 
253 
256 
259 
261 
264 
267 
269 
272 
275 


08 
66 
26 
87 
50 
14 
80 
47 
16 
86 
58 
31 


3254 
3270 
3286 
3302 
3318 
3334 


6 
6 
6 
5 
5 
4 


2257.0 
2266.6 
2276.2 
2285-9 
2295.6 
2305.2 


428 
432 
435 
439 
442 
446 


50 4199 
04 4215 
59 4230 
164246 
75 4261 
35 4277 


8 
3 
8 
3 
8 
3 


53° 

10 
20 
30 
40 
50 


2856 
2867 
2877 
2888 
2898 
2908 


7 
1 
5 

4 
9 


672 
677 
681 
686 
691 
696 


66 
32 
99 
68 
40 
13 


5113 
5128 
5142 
5157 
5172 
5187 


1 

9 
8 
7 
6 


34° 

10 
20 
30 
40 
50 


1751 
1760 
1770 
1779 
1788 
1797 


7 
8 


1 
2 
4 


3350 
3366 
3382 
3398 
3414 
3430 


4 
3 
2 
2 
1 

9 
8 
7 
5 
4 
3 


44° 

10 
20 
30 
40 
50 
45° 
10 
20 
30 
40 
50 


2314-9 
2324.6 
2334.3 
2344.1 
2353.8 
2363-5 
2373-3 
2383-1 
2392-8 
2402-6 
2412-4 
2422-3 


449 
453 
457 
460 
464 
468 


98 4292 
62 4308 
27 4323 
95 4339 
64 4354 
35 4369 


7 
2 
6 

5 
9 


54° 

10 
20 
30 
40 
50 


2919 
2929 
2940 
2951 
2961 
2972 


4 
9 

4 

5 

1 


700 
705 
710 
715 
720 
724 


89 
66 
46 
28 
11 
97 


5202 
5217 
5232 
5246 
5261 
5276 


4 
3 

1 
9 
7 
5 


35° 

10 
20 
30 
40 
50 


1806 
1815 
1824 
1834 
1843 
1852 


6 

7 
9 

1 
3 
5 


278 
280 
283 
286 
289 
292 


05 
82 
60 
39 

20 
02 


3445 
3461 
3477 
3493 
3509 
3525 


472 
475 
479 
483 
487 
490 


08 4385 
82 4400 
59 4416 
37 4431 
16 4446 
984462 


3 
7 

1 
4 
8 
2 


10 
20 
30 
40 
50 


2982 
2993 
3003 
3014 
3025 
3035 


7 
3 
9 
5 
2 
8 


729 
734 
739 
744 
749 
754 


85 
76 
68 
62 
59 
57 


5291 
5306 
5320 
5335 
5350 
5365 


3 

1 
9 
6 
4 

1 


36° 

, 10 
20 
30 
40 
50 


1861 
1870 
1880 
1889 
1898 
1907 


7 
9 
1 
4 
6 
9 


294 
297 
300 
303 
306 
309 


86 

72 
59 
47 
37 
29 


3541 
3557 
3572 
3588 
3604 
3620 


1 

8 
6 
5 
3 


46° 

10 
20 
30 
40 
50 


2432-1 
2441-9 
2451-8 
2461.7 
2471-5 
2481-4 


494 
498 
502 
506 
510 
514 


82 4477 
67,4492 
54*4508 
42 4523 
33 4538 
25 4554 


5 
8 

2 
5 
8 

1 


56° 

10 
20 
30 
40 
50 


3046 
3057 
3067 
3078 
3089 
3100 


5 
2 
9 
7 
4 
2 


759 
764 
769 
774 
779 
784 


58 
61 
66 
73 
83 
94 


5379 
5394 
5409 
5423 
5438 
5453 


a 

5 
2 
9 
6 
3 


37° 

10 
20 
30 
40 
50 


1917 
1926 
1935 
1945 
1954 
1963 


1 
4 
7 

3 
6 


312 
315 
318 

321 
324 
327 


22 
17 
13 
11 
11 
12 


3636 
3651 
3667 
3683 
3699 
3715 


1 
9 
7 
5 
3 



47° 
10 
20 
30 
40 
50 

48° 
10 
20 
30 
40 
50 


2491-3 
2501-2 
2511-2 
2521-1 
2531-1 
2541-0 


518 
522 
526 
530 
534 
538 


20 
16 
13 
13 
15 
18 


4569 
4584 
4599 
4615 
4630 
4645 


4 
7 
9 
2 
4 
7 


57° 
10 
20 
30 
40 
50 


3110 
3121 
3132 
3143 
3154 
3165 


9 

7 
6 
4 
2 
1 


790 
795 
800 
805 
810 
816 


08 
24 
42 
62 
85 
10 


5467 
5482 
5497 
5511 
5526 
5541 


9 
5 

2 
8 

4 



38° 
10 
20 
30 
40 
50 


1972 
1982 
1991 
2000 
2010 
2019 


9 
2 
5 
9 
2 
6 

4 
8 
2 
6 



330 
333 
338 
339 

342 
345 


15 
19 
25 
32 
41 
52 


3730 
3746 
3762 
3778 
3793 
3809 


8 

5 
3 

8 
5 


2551-0 
2561-0 
2571-0 
2581-0 
2591-1 
2601-1 


542 
546 
550 
554 
558 
562 


23 
30 
39 
50 
63 
77 


4660 
4676 
4691 
4706 
4721 
4736 


9 
1 
3 
5 
7 
9 


58° 
10 
20 
30 
40 
50 

59° 

10 
20 
30 
40 
50 


3176 
3186 
3197 
3208 
3219 
3230 



9 
8 
8 

7 
7 


821 
826 
831 
837 
842 
848 


37 
66 
98 
31 
67 
06 


5555 
5570 
5584 
5599 
5613 
5628 
5642 
5657 
5671 
5686 
5700 
5715 


6 
2 
7 
3 
8 
3 


39° 

10 
20 
30 
40 
50 


2029 
2038 
2047 
2057 
2066 
2076 


348 
351 
354 
358 
361 
364 


64 
78 
94 
11 
29 
50 


3825 
3840 
3856 
3872 
3888 
3903 


2 
9 
6 
3 

6 


49° 

10 
20 
30 
40 
50 

50° 
10 
20 
30 
40 
50 

51° 


2611-2 
2621-2 
2631-3 
2641. 4 
2651-5 
2661-6 


566 
571 
575 
579 
583 
588 


94 
12 
32 
54 
78 
04 


4752 
4767 
4782 
4797 
4812 
4827 


1 
3 
4 
5 
7 
8 


3241 
3252 
3263 
3274 
3285 
3296 


7 
7 
7 
8 
8 
9 


853 
858 
864 
869 
875 
880 


46 
89 
34 
82 
32 
84 


8 
3 
8 
3 
8 
2 


40° 

10 
20 
30 
40 
50 


2085 
2094 
2104 
2113 
2123 
2132 


4 
9 
3 
8 
3 
7 


367 
370 
374 
377 
380 
384 


72 
95 
20 
47 
76 
06 


3919 
3935 
3950 
3966 
3981 
3997 


3 

6 
3 
9 
5 


2671-8 
2681-9 
2692-1 
2702. 3 
2712-5 
2722-7 


592 
596 
600 
605 
609 
614 


32 
62 
93 
27 
62 
00 


4842 
4858 
4873 
4888 
4903 
4918 


9 


1 
2 
2 
3 


60° 

10 
20 
30 
40 
50 
61° 


3308 
3319 
3330 
3341 
3352 
3363 



1 
3 
4 
6 
8 


886 
891 
897 
903 
908 
914 


38 
95 
54 
15 
79 
45 


5729 
5744 
5758 
5^772 
5787 
5801 


7 

1 
5 
9 
3 
7 


41° 


2142 


2 


387 


38 


4013 


1 


2732-9 


618 


39 


4933 


4 


3375 





920 


14 


5816 






557 



TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS 

FOR A 1° CURVE. 



A 


T^g' 


Ext. 

Dist. 


Long 
Chord 


A 


T^g- 


Ext. 
Dist. 


Long 
Chord 


A 


T^g- 


Ext. 
Dist. 


Long 
Chord 




E, 


LC, 




E, 


LC. 


81° 


E, 


LC. 


61° 


3375.0 


920.14 


5816. C 


)71° 


4086.9 


1308.2 


6654-4 


4893.6 


1805.3 


7442 - 2 


10' 


3386. 


3 


925. 


85 


5830-4 


t 10 


4099. 


5 


1315. £ 


6668. C 


10 


4908. 


1814. 


7 


7454-9 


20 


3397- 


5 


931. 


58 


5844.'/ 


20 


4112. 


1 


1322. c 


6681 -£ 


20 


4922.5 


1824. 


1 


7467-5 


30 


3408. 


8 


937. 


34 


5859.] 


30 


4124. 


8 


1330. J 


6695-] 


30 


4937.0 


1833- 


6 


7480-2 


40 


3420. 


1 


943. 


12 


5873. < 


40 


4137. 


4 


1337-7 


6708. e 


40 


4951.5 


1843. 


1 


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3431. 


4 


948. 


92 


5887.7 


50 


4150- 


1 


1345.1 


6722-1 


50 


4966-1 


1852. 


6 


7505.4 


^3° 


3442. 


7 


954. 


75 


5902. C 


73° 


4162. 


8 


1352. e 


6735 -e 


82° 


4980-7 


1862. 


2 


7518. 


10 


3454. 


1 


960. 


60 


5916. J 


10 


4175- 


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1360.: 


6749.; 


10 


4995-4 


1871. 


8 


7530.5 


20 


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4 


966. 


48 


5930. £ 


20 


4188- 


4 


1367. e 


6762. £ 


20 


5010-0 


1881. 


5 


7543.1 


30 


3476. 


8 


972. 


39 


5944. J 


30 


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30 


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1891. 


2 


7555.6 


40 


3488. 


2 


978. 


31 


§959. C 


40 


4214- 





1382. £ 


6789.^ 


40 


5039-5 


1900. 


9 


7568.2 


50 


3499. 
3511. 


7 

1 


984. 


27 


5973.3 


50 


4226. 


8 


1390-4 


L6802-f 


_50 
83° 


5054-3 


1910. 


7 


7580-7 
7593-2 


63° 


990. 


24 


5987. £ 


73° 


4239- 


7 


1398. C 


) 6816. £ 


5069-2 


1920. 


5 


10 


3522. 


6 


996. 


24 


6001.7 


10 


4252- 


6 


1405.7 


' 6829 -f 


10 


5084.0 


1930. 


4 


7605.6 


20 


3534. 


1 


1002. 


3 


6015. £ 


20 


4265- 


6 


1413 -f 


) 6843 -C 


20 


5099-0 


1940- 


3 


7618-1 


30 


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1008. 


3 


6030. C 


30 


4278 


5 


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I 6856.^ 


L 30 


5113-9 


1950. 


3 


7630.5 


40 


3557. 


2 


1014. 


4 


6044.2 


40 


4291 


5 


1429- ( 


) 6869-' 


1 40 


5128-9 


1960- 


2 


7643-0 


50 


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7 


1020. 


5 


6058-4 


t 50 


4304 


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3 6896-^ 


L 50 

t 84° 


5143- 9 


1970- 


3 


7655-4 
7667. 8 


64" 


3580. 


3 


1026. 


6 


6072. £ 


)74° 


4317 


6 


1444- ( 


5159.0 


1980- 


4 


10 


3591. 


9 


1032. 


8 


6086.1 


) 10 


4330 


7 


1452- J 


) 6909 ■' 


1 10 


5174-1 


1990- 


5 


7680-1 


20 


3603. 


5 


1039. 





6100.' 


1 20 


4343 


8 


1460-^ 


:6923 ( 


) 20 


5189-3 


2000 


6 7692 5 


30 


3615. 


1 


1045 


2 


6114.1 


1 30 


4356 


9 


1468.^ 


I 6936-: 


I 30 


5204-4 


2010 


8 7704-9 


40 


3626 


8 


1051 


4 


6128 -J 


1 40 


4370 


1 


1476.^ 


. 6949-v 


5 40 


5219-7 


2021 


1 7717-2 


50 


3638 


5 


1057 


7 


6143 -( 
6157. 


) 50 

L 75° 


4383 


3 


1484-^ 
1492-^ 


16962.} 
16976-( 


} 50 

) 85° 


5234-9 


2031 


4 7729-5 


65° 


3650 


2 


1063 


9 


4396 


5 


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2041 


7 


7741 -8^ 


10 


3661 


9 


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2 


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4409 


8 


1500- 


5 6989-: 


2 10 


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2052 


1 


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3673 


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6 


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4423 


1 


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3 7002-^ 


1 20 


5281. 


2062 


5 


7766.3 


30 


3685 


4 


1082 


9 


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2 30 


4436 


4 


1516- 


7 7015- 


3 30 


5296-4 


2073 





7778-6( 


40 


3697 


2 


1089 


3 6213- 


2 40 


4449 


7 


1524- 


3 7028. 


3 40 


5311. 9 


2083 


5 


7790.81 


50 


3709 





1095 


7 


6227- 


2 50 


4463 


1 


1533. 


L 7041. 


3 50 
3 86° 


5327.4 


2094 


1 


7803. 


66° 


3720 


9 


1102 


2 


6241. 


2 76° 


4476 


5 


1541- 


4 7055- 


5343 


2104 


7 


7815.2 


10 


3732 


7 


1108 


6 


6255- 


2 10 


4489 


-9 


1549- 


7 7068- 


2 10 


5358- 6 


2115 


3 


7827.4 


20 


3744 


6 


1115 


1 


6269. 


L 20 


4503 


-4 


1558- 


7081. 


3 20 


5374-2 


2126 





7839.6 


30 


3756 


5 


1121 


7 


6283- 


L 30 


4516 


-9 


1566- 


3 7094. 


4 30 


5389-9 


2136 


7 


7851.7 


40 


3768 


5 


1128 


2 


6297. 


D 40 


4530 


-4 


1574- 


7 7107- 


5 40 


5405-6 


2147 


5 


7863-8 


50 


3780 


4 


1134 


8 


6310. 


9 50 


4544 





1583- 


1 7120. 


5 50 


5421-4 


2158 


4 


7876.0 


67° 


3792 


4 


1141 


4 


6324- 


8 77° 


4557 


-6 


1591. 


6 7133. 


6 87° 


5437-2 


2169 


.2 


7888.1 


10 


3804 


4 


1148 





6338- 


7 10 


4571 


-2 


1600. 


1 7146- 


3 10 


5453-1 


2180 


-2 


7900.1 


20 


3816 


4 


1154 


7 


6352- 


3 20 


4584 


• 8 


1608. 


6 7159- 


6 20 


5469-0 


2191 


-1 


7912.2 


30 


3828 


4 


1161 


3 


6366- 


4 30 


4598 


-5 


1617- 


1 7172. 


6 30 


5484-9 


2202 


-2 


7924.3 


40 


3840 


5 


1168 


1 


6380. 


3 40 


4612 


-2 


1625- 


7 7185- 


6 40 


55009 


2213 


-2 


7936.3 


50 


3852 


6 


1174 


8 


6394- 


1 50 

78° 


4626 


-0 


1634. 


4 7198- 


6 50 

6 88° 


55170 
5533-1 


2224 
2235 


-3 

-5 


7948.3 
7960.3 


68° 


3864 


7 


1181 


.6 


6408- 


4639 


-8 


1643- 


7211- 


10 


3876 


8 


1188 


4 


6421- 


} 10 


4653 


6 


1651- 


7 7224- 


5 10 


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2246 


•7 


7972.3 


20 


3889 


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1195 


.2 


6435- 


3 20 


4667 


-4 


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5 7237. 


4 20 


5565--^ 


2258 


-0 


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30 


3901 


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1202 


.0 


6449- 


4 30 


4681 


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2 7250. 


4 30 


5581-6 


2269 


-3 


7996.2 


40 


3913 


4 


1208 


9 


6463- 


1 40 


4695 


-2 


1678- 


1 7263. 


3 40 


5597-J 


2280 


• 6 


8008.1 


50 


3925 


• 6 


1215 


8 


6476- 


9 50 
6 79° 


4709 


-2 


1686- 


9 7276. 


1 50 


5614-2 


2292 





8020. C 
8031.8 


69° 


3937 


.9 


1222 


7 


6490- 


4723 


-2 


1695- 


8 7289- 


89° 


5630-5 


2303 


-5 


10 


3950 


.2 


1229 


.7 


6504- 


4 10 


4737 


-2 


1704- 


7 7301- 


9 10 


5646- £ 


2315 


-0 


8043.8 


20 


3962 


.5 


1236 


.7 


6518- 


1 20 


4751 


.2 


1713- 


7 7314- 


7 20 


5663-4 


2326 


-6 


8055.7 


30 


3974 


.8 


1243 


.7 


6531- 


J 30 


4765 


3 


1722. 


7 7327- 


5 30 


5679-9 


2338 


-2 


8067. £ 


40 


3987 


.2 


1250 


8 


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5 40 


4779 


.4 


1731- 


7 7340- 


3 40 


5696-4 


2349 


-8 


8079.3 


50 


3999 


.5 


1257 


9 


6559- 


1 50 
8 80° 


4793 


-6 


1740- 


8 7353- 

9 7365- 


1 50 
9 90° 


5713-0 


2361 


-5 


8091. 2 


70° 


4011 


.9 


1265 


.0 


6572. 


4808 


-7 


1749- 


5729-7 


2373 


-3 


8103. C 


10 


4024 


.4 


1272 


.1 


6586. 


4 10 


4822 


.0 


1759- 


7378- 


7 10 


5746-3 


2385 


-1 


8114.7 


20 


4036 


.8 


1279 


.3 


6600. 


L 20 


4836 


-2 


1768- 


2 7391- 


4 20 


5763-1 


2397 





8126. e 


30 


4049 


■ 3 


1286 


.5 


6613. 


7 30 


4850 


-5 


1777. 


4 7404- 


L 30 


5779-9 


2408 


9 


8138-2 


40 


4061 


.8 


1293 


7 


6627. 


3 40 


4864 


8 


1786. 


7 7416- 


\ 40 


5796-7 


2420 


9 


8150. C 


50 


4074 


.4 


1300 


9 


6640. 


9 50 


4879 


2 


1796. 


[) 7429 . 


5 50 


5813-6 
5830. 5 


2432 


9 


8161-7 


TJLl 


4086 


• 9 


1308 


• 2 


6654. 


4 81° 


4893 


6 


1805. 


3 7442.. 


2 91° 


2444 


9 


8173.4 



558 



TABLE III.— SWITCH LEADS AND DISTANCES 
LEAD-RAILS CIRCULAR THROUGHOUT; GAUG E 4' s¥'. See § 262. 



Frog 
No. 
(n). 



4 

4 

5 

5 

6 

6 

7 

7 

8 

8 

9 

9 

10 

10 

11 

11 

12 



Frog Angle 



14° 15' 00'' 
12 40 49 
11 25 
10 23 



16 
20 



8 47 
8 10 



31 38 
51 



16 

7 37 41 

7 09 10 

6 43 59 

6 21 35 

6 01 32 
43 



Lead (L) 
(Eq. 79) 



5 

5 

5 

4 58 45 

4 46 19 



27 09 
12 18 



37 
42 
47 
51 
56 
61 
65 
70 
75 
80 
84 
89 
94 
98 
103 
108 
113 



• 67 
•37 

• 08 
.79 

• 50 

• 21 

• 92 

• 62 

• 33 

• 04 
.75 

46 

• 17 
87 
58 

• 29 

• 00 



' Chord 
(QT) 
(Eq. 77). 



Radius of 
Lead-rails 
(r,Eq. 78). 



3738 
42^12 
46.85 
51^58 
5630 
61. 03 
65-75 
70-47 
75-19 
79-90 
84-62 
8933 
94-05 
98-76 

iai-47 

108- 19 
112-90 



Log r. 



150 

190 

235 

284 

339 

397 

461 

529 

602 

680 

762 

849 

941 

1038 

1139 

1245 

1356 



-67 
-69 
-42 

• 85 

• 00 
-85 

• 42 
-69 
•67 

36 
•75 

• 85 

• 67 
•19 

• 42 
■ 36 

• 00 



Length of 
Switch-rails 
(QK, eq.81) 



17801 
•28032 
•37183 
.45462 
■53020 
•59972 
.66409 
. 72402 
.78007 
.83273 

88238 
.92934 
•97389 
•01627 
•05668 
•09529 
.13226 



TURNOUTS WITH 
R 



Frog 
No. 
(n). 



4 
4.5 



5 

5 

6 

6 

7 

7 

8 

8 

9 

9 

10 

10 

11 

11 

12 



I Switch 

Point 

Angle 

(a). 



3° 40' 
3 40 
45 
45 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 



L'gth 

of 
Switch 
Point 

(DN). 



STRAIGHT POINT-RAILS 
AILS; GAUGE 4' 8i" . See § 



11 
13 

14 
16 
17 
19 
20 
22 
23 
24 
26 
27 
29 
30 
32 
33 
35 



•73 
19 
•65 
.15 
•64 

• 09 
.53 
•03 
.48 
.93 
.43 

97 

• 37 

• 85 
•31 

78 
•17 



Frog 
No. 



4 

4.5 

5 

5.5 

6 

6.5 

7 

7.5 

8 

85 

9 

9.5 
10 

10^5 
11 

11^5 
12 



AND STRAIGirr FROG- 
265. 



L'gth 

of 
Str'g't 

Frog- 
rail (/). 



150 

1^69 

187 

2-06 

2-25 

2-44 

2-62 

2-81 

3-00 

3. 19 

37 

56 

75 

94 

12 

31 

50 



Lead 

(L) 
(Eq. 
90). 



-20 
-29 

• 85 

• 16 

• 00 

• 84 

• 65 

• 36 

• 04 
-60 

20 
-70 
-04 
-51 
-82 
-09 
-16 



Chord 
(ST) 
(Eq. 
88). 



Radius of 
Lead- 
rails 
(r, Eq 
87). 



125 
159 
197 
240 
288 
340 
397 
460 
527 
600 
681 
767 
858 
959 
1065 
1180 
1299 



-21 


2. 


• 25 




• 65 




• 44 




• 09 




• 19 




• 65 




• 00 




• 91 




.94 




• 16 




• 11 




• 14 




• 00 


2. 


• 52 


3. 


• 16 


3^ 


.93 


3. 



Log r. 



2. 09764 

.20208 

.29589 

.38100 

.45953 

•53172 

•59950 

•66276 

•72256 

•77883 

•83325 

88486 

93356 

2 98182 

J02756 

J07194 

M1392 



Degree 

of 
Curve 

(d). 



05' 

36 

22 

00 

59 

54 

27 

29 

52 

33 

25 

28 

41 

59 

23 

51 

24 



Frog 

No. 
(n). 



Frog 

No. 
(n). 



TRIGONOMETRICAL FUNCTIONS OF THE FROG ANGLES (F). 



4 

4.5 

5 

5.5 

6 

6.5 

7 

7.5 

8 

85 

9 

9.5 
10 

10.5 
11 

11.5 
12 



4 

4.5 

5 

5.5 

6 

6.5 

7 

7.5 

8 

85 

9 

9.5 
10 

10. 5 
11 

11. 5 
12 



Frog Angle 

(F). 



14° 15' 00' 

12 40 49 

11 25 16 

10 23 20 

9 31 38 

8 47 51 

8 10 16 

7 37 41 

7 09 10 

6 43 59 

6 21 35 

6 01 32 

5 43 29 

5 27 09 

5 12 18 

4 58 45 

4 46 19 



Nat. 
sin F. 



•24615 
•21951 
19802 
•18033 
•16552 
•15294 
.14213 
•13274 
•12452 
•11724 
•11077 
•10497 
.09975 
•09502 
•09072 
•08679 
•08319 



Nat. 
cos F. 



.96923 
.97561 
•98020 
•98360 
■98621 
98823 
98985 
•99115 
•99222 
•99310 
•99385 
.99448 
•99501 
•99548 
•99588 
•99623 
•99653 



Log 
sin F. 



)-39120 
•34145 
•29670 
•25606 
•21884 
•18453 
■15268 
■12301 
■09522 
•06909 
• 04442 

)• 02107 

399891 
•97781 
•95770 
•93848 

3 92007 



Log 
cos F. 



9.98642 
•98927 
.99131 
•99282 
•99397 
•99486 
•99557 
•99614 
•99660 
•99699 
•99732 
•99759 
•99783 
.99803 
■99820 
.99836 

9. 99849 



Log 
cot F. 



10^59522 
•64782 
•69461 
•73675 
■77513 
■81033 
■84288 
■87313 
•90138 
•92790 
•95289 
•97652 
10-99892 
1102021 
•04050 
•05987 
07842 



11 



Log 
vers F. 



•48811 
•38721 
•29670 
•21467 
•13966 
•07058 
00655 
•94691 
•89110 
•83864 
•78915 
•74232 
•69788 
•65560 
.61528 
57676 
53986 



Frog 
No. 
(n). 



4 

4.5 

5 

5.5 

6 

6.5 

7 

7.5 

8 

85 

9 

95 
10 

10.5 
11 
11. 
12 



559 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 



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CO rH T*< rH Tl< rH CO 


o 






o 


J*_ 


CO 


CO CO OQ CM iH rH O 


o 






<M 


o 


t- lo <M o t^ m 


o 


o 






Tt< 


CSI 


CO 


O iH in O CO rj< 


o 


CO 




Oi 


^ 
















00 


•«* 


t* 


oo CD iH in in CO 


o 


t«. 






CO 


CM 


o 


•* CM O CO O CO 


o 


CO 


I 




o 
CO 


CO 


CO 


<N Cq <N iH rH O 


o 


o 




^ 












"ee 




Csj 


in 


!>■ 


o CM in c>. o 


o 


in CM 


75 




CS] 


1— 1 


CO 


00 lo r^ o o 


o 


<:*< CSI 




X 


^ 
















■^ 


1— 1 


in 


t^ CO CO 00 o 


o 


CO 03 


c 




in 


>* 


CM 


o •«* CM in CO 


o 


CO o 


s 

3 


o 


<N 


CSI 


CM iH rH O O 


o 


O iH 




^ 






J 






o 


t"- 


in 


C<J O t^ IC3 


O 


o CM in 






o 


o 


•«^ 


lO CO CO rH 


o 


O in rH 


.E 


i>« 


^ 
















in 


CO 


00 


iH CM O CO 


o 


O rH CD 


o 




pH 


o 


•^ 


CO rH in CM 


o 


CO O CO 


X, 




o 












c 




cq 


<N 


1—1 


iH iH O O 


o 


O rH rH 












^ 
^ 




V. 


C 


C<1 


in c>- o 


o 


in CO o t- 




00 


o 


in 


iH O CO 


o 


rH CM O O 


;o 














T 




o 


o 


CD 


iH CO <M 


o 


CD -«;t< in 00 


<U 




•^ 


CO 


1—1 


O "«* oq 


o 


CM m CQ in 


*c. 


J o_^ 












p 

o 








iH O O 


OfcO O rH rH 




:: 












o 

c 




in 


CM 


O 


D- in 


O 


o CQ in j>. o 




(—1 


in 


O 


CO •* 


O 


CO in tjh o o 


-^:: 


W5 


.^ 




















o 


in oo 


O 


05 CO CO CO in 


'c 




1— 1 


o 


in 


CO rH 


O 


(M -"i* rH ^ rH 


c. 




o 
















i-H 


I-H 


o 


O O 


O 


O O rH rH 


CSI_ 




^ 












•+J 




<N 


in 


t- 


O 


o 


in cq o i>- in cq 






in 


r^ 


o 


o 


o 


'^ eg CO o iH in 


C3 


Tt^ 


^ 
















CD 


00 


00 


in 


o 


00 03 (M 00 CD CO 


c 




■^ 


CO 


CM 




o 


n-1 CO o csj in cq 


o 


i O 












h: J o 


o 


o 


c 


o 


O O rH rH iH (N 










c3 














i ^ 


t>- 


in 


O 


o eg in t* o csj in 


43 


1 CO 


co 




O 


O lO iH O CO <N ^ 


^ 


cc 












t>- 


o 




o 


in iH iH CO t"- ^ CO 


;:: 


<N 


CM 




o 


rH CO in iH CO O CO 




o 
O 


o 


o 


o 


O O O iH nH CM (M 




:: 












t^ 


o 


c 


locsiot-incaot* 


c 




o 


CO 


c 


iHCMOOTt^incOCO 


c 


W 


. 










i ^ 


t> 


c 


.H^OOOOOrHt-in 


t 


1 rH 


o 


c 


iHCsi^iniHTi<oco 


o 




o 








q3 




O 


o 


c 


OOOOiHiHCMCM 




^ 










in 


o 


o 


CMir3t«»OC3iOI>-0 






rf 


o 


CO 


m'^oocgrHooeo 




^ 














cc 


o 


tx 


cDoocooo)rHineq 

iHCM^OiHTi^OCO 




ly 


c 


o 


OOOr-HrHrHCSJCS 


1 "b 


in 


(N 


ot-.incMor>.ino 






■<^ 


CM 


COCrHinOCOT*<Cq 




O lo 


CO 


03 


t^ooiHCDininoO'^ 












i-iCviT^mrHcomcM 


I 


O 


o 


.2. 


OOOOrHrHrHCM 


bi 










.s 




















X 


O- 


tH 


W 


CC-^WiCOf^QCCi© 


bl 








tH 




p 

























560 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 
Dimensions of various 0° 30'-per-25-feet spirals. — Part B. 



Deg. 

of 


L'gth 
of 


ZK 


QK 


A'N 


NQ 


A' toZ. 


^^f 








curve. 


spiral 










Defl. 


Dist. 


curve. 


1° 00' 


25 


0.03 


25.00 


0.01 


12.50 


0° 


04' 


12.50 


1° 00' 


10 


50 


0.14 


50.00 


0.03 


17.86 





11 


32.14 


10 


20 


50 


0.14 


50.00 


0.04 


21.88 





11 


28.13 


20 


30 


50 


0.14 


50.00 


0.05 


25.00 





11 


25.00 


30 


40 


75 


0.38 


75.00 


0.09 


30.00 





22i 


45.00 


40 


50 


75 


0.38 


75.00 


0.11 


43 . 09 





22^ 


40.91 


50 


2** 00' 


75 


0.38 


75.00 


0.14 


37.50 


0° 


22r 


37.50 


2° 00' 


10 


100 


0.82 


100.00 


0.19 


42.30 





37^ 


57.69 


10 


20 


100 


0.82 


100.00 


0.23 


46.43 





37^ 


53.57 


20 


30 


100 


0.82 


100.00 


0.27 


50.00 





37| 


50 00 


30 


40 


125 


1.50 


124.99 


0.35 


54.68 





56 


7C.31 


40 


60 


125 


1.50 


124.99 


0.42 


58.81 





56 


66.18 


50 


8«> 00' 


125 


1.50 


124.99 


0.48 


62.49 


0° 


56' 


62.50 


3° 00' 


10 


150 


2.48 


149 97 


0.58 


67.09 




19 


82.59 


10 


20 


150 


2.48 


149.97 


0.68 


71.24 




19 


78.75 


20 


30 


150 


2.48 


149.97 


0.76 


74.98 




19 


75.00 


30 


40 


175 


3.82 


174.93 


0.90 


79.52 




45 


95.45 


40 


50 


175 


3.82 


174.93 


1.03 


83.67 




45 


91.30 


50 


4*> 00' 


175 


3.82 


174.93 


1.15 


87.47 


1° 


45' 


87.50 


4° 00' 


10 


200 


5.56 


199.87 


1.32 


91.96 


2 


15 


108.00 


10 


20 


200 


5.56 


199.87 


1.48 


96.11 


2 


15 


103.85 


20 


30 


200 


5.56 


199.87 


1.63 


99.95 


2 


15 


100.00 


30 


40 


225 


7.79 


224.77 


1.88 


104.40 


2 


49 


120.54 


40 


50 


225 


7.79 


224.77 


2.08 


108.55 


2 


49 


116.38 


50 


6° 00' 


225 


7.79 


224.77 


2.27 


112.42 


2° 


49' 


112.50 


5" 00' 


10 


250 


10.49 


249.62 


2.51 


116.83 


3 


26 


133.06 


10 


20 


250 


10.49 


249.62 


2.76 


120.98 


3 


26 


128.91 


20 


30 


250 


10.49 


249.62 


3.00 


124.88 


3 


26 


125.00 


SO 



561 



TABLE lY.— ELEMENTS OF TRANSITION CURVES. 

Dimensions of various 0° 30'-per-25-feet spirals. — Part C. 

Values of AN. 









Degree of curve. 






A 














r 


2° 


3° 


4° 


5° 


5° 30' 


2° 


0.00 


0.00 


0.01 


0.02 


0.04 


0.05 


4 


0.00 


0.01 


0.02 


0.04 


0.08 


0.10 


6 


0.00 


0.01 


0.03 


0.06 


0.12 


0.16 


8 


0.00 


0.01 


0.03 


0.08 


0.16 


0.21 


10 


0.00 


0.01 


0.04 


0.10 


0.20 


0.26 


12° 


0.00 


0.01 


0.05 


0.12 


0.24 


0.32 


14 


0.00 


02 


0.06 


0.14 


0.28 


0.37 


16 


00 


0.02 


0.07 


16 


0.32 


0.42 


18 


0.00 


0.02 


0.08 


18 


0.36 


47 


20 


0.00 


0.02 


0.08 


0.20 


0.40 


0.53 


22° 


0.00 


0.03 


0.09 


0.22 


0.44 


0.58 


24 


0.00 


03 


0.10 


0.24 


0.48 


0.64 


26 


0.00 


03 


0.11 


0.26 


0.52 


0.69 


28 


0.00 


0.03 


12 


0.29 


0.57 


0.75 


30 


0.00 


0.04 


0.13 


0.31 


0.61 


0.80 


32° 


0.00 


0.04 


0.14 


0.33 


0.65 


0.86 


34 


0.00 


0.04 


0.15 


0.35 


0.69 


0.92 


36 


0.00 


0.04 


0.16 


37 


0.74 


0.97 


38 


0.00 


05 


0.16 


39 


0.78 


1.03 


40 


0.00 


0.05 


0.17 


0.42 


0.83 


1.09 


42° 


0.00 


05 


0.18 


0.44 


0.87 


1.15 


44 


0.00 


0.06 


0.19 


0.46 


0.92 


1.21 


46 


0.00 


0.06 


20 


0.49 


0.96 


1.27 


48 


0.00 


0.06 


21 


0.51 


1.01 


1.34 


50 


0.00 


0.06 


0.22 


0.53 


1.06 


1.40 


52° 


0.00 


0.07 


0.23 


0.56 


1.11 


1.46 


54 


0.01 


0.07 


0.24 


0.58 


1.16 


1.53 


56 


0.01 


0.07 


0.25 


0.61 


1.21 


1.59 


58 


0.01 


0.08 


0.26 


0.64 


1.26 


1.66 


60 


0.01 


0.08 


0.28 


0.66 


1.31 


1.73 


62° 


0.01 


0.08 


0.29 


0.69 


1.36 


1.80 


64 


0.01 


09 


30 


0.72 


1.42 


1.87 


66 


0.01 


0.09 


0.31 


0.74 


1.47 


1.95 


68 


01 


0.09 


0.32 


0.77 


1.53 


2.02 


70 


0.01 


0.10 


0.33 


0.80 


1.59 


2.10 


72° 


0.01 


0.10 


0.35 


0.83 


1.65 


2.18 


74 


0.01 


0.10 


0.36 


0.83 


1.71 


2.26 


76 


0.01 


0.11 


0.37 


0.90 


1.77 


2.34 


78 


0.01 


0.11 


0.39 


0.93 


1.84 


2 43 


80 


0.01 


0.11 


0.40 


0.96 


1.91 


2.51 


82° 


0.01 


0.12 


0.42 


1.00 


1.97 


2.60 


84 


0.01 


0.12 


0.43 


1.03 


2.04 


2.70 


86 


0.01 


0.13 


0.45 


1.07 


2.12 


2.79 


88 


0.01 


0.13 


0.46 


1.10 


2.19 


2.89 


90 


0.01 


0.14 


0.48 


1.15 


2.27 


3.00 


92° 


0.01 


0.14 


0.49 


1.19 


2.35 


3.10 


94 


0.01 


0.15 


0.51 


1.23 


2.44 


3.21 


96 


0.01 


0.15 


0.53 


1.27 


2.52 


3.33 


98 


0.01 


0.16 


0.55 


1.32 


2.61 


3.45 


100 


0.01 


0.16 


0.57 


1.37 


2.71 


3.57 



662 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 



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CD 


in 


00 


c>. 


g 




Id 


■<d< 


CI 


o 


CO 


O ".iH 


cq 


rH 


o 


o 


c 




o 

O 


o 


o 


o 


o 


rH iH 


cq 


CO 


■^ 


in 


«*-( 




:: 




















uTJ 




ID 


o 


o 


o 


in 


o in 


o 


in 


00 


iH 


fl 




rH 


o 


o 


CO 


•^ 


O rH 


CO 


^ 


in 


t-i 


o 


w 


^ 




















■■+i 


CD 


in 


o 


cq 


CO 


O CO 


t>. 


CO 


■«*< 


rH 


o 

^ 




CM 


rH 


o 


cq 


Ti* 


CM in 


co 


cq 




i—i 




o 




















(D 




O 


o 


o 


o 


o 


t-^ iH 


cq 


CO 


■«d< 


in 


Q 




:; 






















o 


o 


o in 


o 


in o 


in 


o 


CO 


in 






CO 


o 


o •* 


CO 


rH O 


'«1< 


CO 




in 




^ 
























I> 


o 


in CO 


t- 


CD O 


00 


cq 




•^ 






o 


o 


rH CO 


to 


cq o 


CO 


cq 


'-' 


o 






o 

o 


o 


o o 


o 


rH cq 


cq 


CO 


-^ 


in 






o 


o 


in o 


in 


o in 


o 


CO 


CD 


00 








CO 


^ O 


rH 


CO Ti< 


o 


rH 


cq 


CO 




C^ 


o 


t> 


00 in 


CO 


cq CO 


o 


rH 


c^ 


00 








o 


rH CO 


in 


cq in 


CO 


'-' 


in 


rJ4 






o 

o 


o 


o o 


o 


,-i r-i 


cq 


CO 


CO 


■^ 


Wi 




















c 








































X 


c^ 


H 


N W 


^ 


US ;d 


l^ 


QC 


Oi 


O 


M 


















iH 


•P +3 






















'^ 

























5as 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 
Dimensions of various l°-per-25-feet spirals. — Part B. 



^o?- 


L'gth 
of 


ZK 


QK 


A'N 


NQ 


A' to Z. 


Deg. 
of 








curve. 


spiral 










Defl. 


Dist. 


curve. 


2^ 00' 


25 


0.06 


25.00 


0.03 


12.50 


0° 


07^ 


12.50 


2° 00' 


10 


50 


0.27 


50.00 


0.05 


15.38 





22^ 


34.62 


10 


20 


50 


0.27 


50.00 ; 


0.06 


17.86 





22^ 


32.14 


20 


30 


50 


0.27 


50.00 1 


0.08 


20.00 





22^ 


30.00 


30 


40 


50 


0.27 


50.00 


0.09 


21.87 





22i 


28.13 


40 


50 


50 


0.27 


50.00 


0.10 


23.53 





22^ 


26.17 


50 


8** 00' 


50 


0.27 


50.00 


0.11 


25.00 


0° 


22i' 


25.00 


3° 00' 


10 


75 


0.76 


74.99 


0.14 


27.63 





45 


47.37 


10 


20 


75 


0.76 


74.99 


0.17 


29.99 





45 


45.00 


20 


30 


75 


0.76 


74.99 


0.20 


32.13 





45 


42.86 


30 


40 


75 


0.76 


74.99 


0.23 


34.08 





45 


40.19 


40 


50 


75 


0.76 


74.99 


0.25 


35.86 





45 


39.13 


50 


4° 00' 


75 


0.76 


74.99 


0.27 


37.49 


0° 


45' 


37.50 


4° 00' 


10 


100 


1.64 


99.98 


33 


39 98 




15 


60.00 


10 


20 


100 


1.64 


99.98 


38 


42.29 




15 


57.69 


20 


30 


100 


1.64 


99 98 


42 


44.43 




15 


55.56 


30 


40 


100 


1.64 


99.98 


47 


46.41 




15 


53.57 


40 


50 


100 


1.64 


99 98 


0.51 


48.26 




15 


51.72 


50 


6° 00' 


100 


1.64 


99 98 


55 


49.98 


10 


15' 


50.00 


5° 00' 


10 


125 


3.00 


124.94 


62 


52.39 




52^ 


72 58 


10 


20 


125 


3.00 


124.94 


0.70 


54.65 




52i 


70.31 


20 


30 


125 


3.00 


124.94 


0.77 


56.78 




52.V 


68.18 


30 


40 


125 


3.00 


124.94 


83 


58.79 




52^ 


66.18 


40 


50 


125 


3.00 


124.94 


0.90 


60.67 




52j 


64.29 


50 


e«» 00' 


125 


3.00 


124.94 


0.95 


62.46 


V 


52 i' 


62.50 


6° 00' 


10 


150 


4.96 


149.87 


1.06 


64.81 


2 


37^ 


85.14 


10 


20 


150 


4.96 


149 87 


1.16 


67 04 


2 


37i, 


82.89 


20 


30 


150 


4.96 


149 87 


1.26 


69 17 


2 


37^1 


80.77 


30 


40 


150 


4.96 


149 87 


1.35 


71 18 


2 


37^ 


78.75 


40 


50 


150 


4.96 


149.87 


1.44 


73.10 


2 


37i 


76.83 


50 


7^ 00' 


150 


4.96 


149.87 


1.52 


74.92 


2° 


37*' 


75.00 


7° 00' 


10 


175 


7.63 


174.72 


1.67 


77.23 


3 


30 


97.67 


10 


20 


175 


7.63 


174.72 


1.80 


79.44 


3 


30 


95.45 


20 


30 


175 


7.63 


174.72 


1.93 


81.55 


3 


30 


93.33 


30 


40 


175 


7.63 


174.72 


2.05 


83.58 


3 


30 


91.30 


40 


50 


175 


7.63 


174.72 


2.17 


85.51 


3 


30 


89.36 


50 


8«> 00' 


175 


7.63 


174.72 


2.29 


87.37 


3° 


30' 


87.50 


8° 00' 


10 


200 


11.11 


199.48 


2.46 


89.64 


4 


30 


110.20 


10 


20 


200 


11.11 


199.48 


2.64 


91.83 


4 


30 


108.00 


20 


30 


200 


11.11 


199.48 


2.80 


93 94 


4 


30 


105.88 


30 


40 


200 


11.11 


199.48 


2.96 


95.96 


4 


30 


103 85 


40 


50 


200 


11.11 


199.48 


3.10 


97.91 


4 


30 


101.89 


50 


9° 00' 


200 


11.11 


199.48 


3.26 


99.79 


4° 


30' 


100.00 


9° 00' 


10 


225 


15.50 


224.09 


3.48 


102.03 


5 


37^ 


122.73 


10 


20 


225 


15.50 


224.09 


3.69 


104.20 


5 


37^ 


120.54 


20 


30 


225 


15.50 


224.09 


3.90 


106.29 


5 


37* 


118.42 


30 


40 


225 


15.50 


224.09 


4.10 


108 32 


5 


37* 


116.38 


40 


50 


225 


15.50 


224.09 


4.29 


110.28 


5 


37* 


114.41 


50 


10*» 00' 


225 


15.50 


224.09 


4.48 


112.17 


5° 


37*' 


112.50 


10° 00' 


10 


250 


20.91 


248 . 50 


4.74 


114.37 


6 


52* 


135.25 


10 


20 


250 


20.91 


248 . 50 


5.00 


116 53 


6 


52* 


133.06 


20 


30 


250 


20.91 


248 . 50 


5.25 


118 62 


6 


52* 


130.95 


30 


40 


250 


20.91 


248 . 50 


5.50 


120.64 


6 


52* 


128.91 


40 


50 


250 


20.91 


248 . 50 


5.73 


122.60 


6 


52* 


126.92 


50 


ir 00' 


250 


20.91 


248 . 50 


5.96 


124.50 


6° 


52*' 


125.00 


11° 00' 



564 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 

Dimensions of various l°-per-25-feet spirals. — Part C. 

Values of ^A^. 





Degree 


of curve. 








A 


































2° 


3° 


4° 


5'' 6° 


T 


8° 


9° 


10° 


11° 


2 


0.00 


0.00 


0.00 


0.01 0.02 


0.03 


0.04 


0.06 


0.08 i 


0.11 


4 


0.00 


0.00 


0.01 


0.02 


0.03 


0.05 


0.08 


0.11 


0.16 


0.21 


6 


0.00 


0.01 


0.01 


0.03 


0.05 


0.08 


0.12 


0.17 


0.23 


0.31 


8 


0.00 


0.01 


0.02 


0.04 


0.07 


0.11 


0.16 


0.23 


0.31 


0.42 


10 


0.00 


0.01 


0.02 


0.05 


0.08 


0.13 


0.20 


0.29 


0.39 


0.52 


12 


0.00 


0.01 


0.03 


0.06 


0.10 


0.16 


0.24 


0.34 


0.47 


0.63 


14 


0.00 


01 


0.03 


0.07 


0.12 


0.19 


0.28 


0.40 


0.55 


0.73 


16 


0.00 0.02! 


0.04 


0.08 


0.13 


0.21 


0.32 


0.46 


0.63 


0.84 


18 


0.00 0.02 


0.04 


0.09 


0.15 


0.24 


0.36 


0.52 


0.71 


0.94 


20 


0.00 0.02 


0.05 


0.10 


0.17 


0.27 


0.40 


0.57 


0.79 


1.05 


22 


0.01 


0.02 


0.05 


0.11 


0.19 


0.30 


0.44 


63 


0.87 


1.16 


24 


0.01 


0.02 


0.06 


0.12 


0.20 


0.32 


0.49 


0.69 


0.95 


1.27 


26 


0.01 


0.03 


0.06 


0.13 


0.22 


0.35 


0.53 


0.75 


1.03 


1.38 


28 


0.01 


0.03 


0.07 


0.14 


0.24 


0.38 


0.57 


0.81 


1.11 


1.49 


30 


0.01 


0.03 


0.07 


0.15 


0.26 


0.41 


0.61 


0.87 


1.20 


1.60 


32 


0.01 


0.03 


0.08 


0.16 


0.27 


0.44 


0.65 


0.93 


1.28 


1.71 


34 


0.01 


0.03 


0.08 


17 


0.29 


0.47 


0.70 


1.00 


1.37 


1.82 


36 


01 


0.04 


0.09 


18 


31 


0.49 


0.74 


1.06 


1.45 


1.94 


38 


01 


0.04 


0.09 


19 


33 


0.52 


0.79 


1.12 


1.54 


2.05 


40 


01 


0.04 


0.10 


0.20 


0.35 


0.55 


0.83 


1.19 


1.63 


2.17 


42 


0.01 


0.04 


0.10 


0.21 


0.37 


0.58 


0.88 


1.25 


1.72 


2.28 


44 


0.01 


04 


0.11 


0.22 


0.38 


0.62 


0.92 


1.32 


1.81 


2.41 


46 


0.01 


0.05 


0.12 


0.23 


40 


0.65 


0.97 


1.38 


1.90 


2.53 


48 


0.01 


0.05 


0.12 


0.24 


0.42 


0.68 


1.02 


1.45 


1.99 


2.65 


50 


0.01 


0.05 


0.13 


0.25 


0.44 


0.71 


1.07 


1.52 


2.09 


2.78 


52 


0.01 


0.05 


0.13 


0.27 


0.46 


0.74 


1.11 


1.59 


2.18 


2.91 


54 


0.01 


0.06 


0.14 


0.28 


49 


0.78 


1.16 


1.66 


2.28 


3.04 


56 


01 


06 


0.14 


0.29 


0.51 


0.81 


1.21 


1.74 


2.38 


3.17 


58 


0.02 


06 


0.15 


0.30 


53 


0.85 


1.27 


1.81 


2.48 


3.31 


60 


0.02 


0.06 


0.16 


0.31 


0.55 


0.88 


1.32 


1.88 


2.58 


3.44 


62 


0.02 


0.07 


0.16 


33 


0.57 


0.92 


1.37 


1.96 


2.69 


3.58 


64 


0.02 


0.07 


0.17 


0.34 


60 


0.95 


1.43 


2.04 


2.80 


3.73 


66 


0.02 


0.07 


0.18 


0.35 


0.62 


0.99 


1 1.48 


2.12 


2.91 


3.87 


68 


0.02 


0.07 


0.18 


0.37 


0,64 


1.03 


1.54 


2.20 


3.02 


4.02 


70 


0.02 


0.08 


0.19 


0.38 


0.67 


1.07 


1.60 


2.28 


3.13 


4.18 


72 


0.02 


0.08 


0.20 


0.40 


0.69 


1.11 


1.66 


2.37 


3.25 


4.33 


74 


0.02 


0.08 


0.20 


0.41 


0.72 


1.15 


1.72 


2.46 


3.38 


4.49 


76 


0.02 


0.09 i 0.21 


0.43 


0.74 


1.19 


1.79 


2.55 


3.50 


4.66 


78 


0.02 


0.09 


0.22 


0.44 


77 


1.23 


' 1.85 


2.64 


3.63 


4.83 


80 


0.02 


0.09 


0.23 


0.46 


0.80 


1.28 


j 1.92 


2.74 


3.76 


5.00 


82 


0.02 


09 


0.24 


0.47 


83 


1.32 


i 1.99 


2.83 


3 89 


5.18 


84 


0.03 


0.10 


0.24 


0.49 


86 


1 37 


2.06 


2.94 


4.03 


5.37 


86 


0.03 


0.10 


0.25 


0.51 


0.89 


1.42 


'2.13 


3.04 


4.18 


5.56 


88 


0.03 


0.11 


0.26 


0.53 


92 


1.47 


2.21 


3.15 


4.33 


5.76 


90 


0.03 


0.11 


0.27 


0.55 


0.95 


1.52 


2.29 


3.26 


4.48 


5.96 


92 


0.03 


0.11 


0.28 


0.56 


99 


1.58 


2.37 


3.38 


4.64 


6.17 


94 


0.03 


0.12 


0.29 


0.58 


1.02 


1.63 


2.45 


3 50 


4.80 


6.39 


96 


0.03 


0.12 


0.30 


61 


1.06 


1.69 


2.54 


3.62 


4.97 


6.62 


98 


0.03 


0.13 


0.31 


0.63 


1.10 


1.75 


2.63 


3.75 


5.15 


6.86 


100 


0.03 


0.13 


0.32 


0.65 


1.13 


1.82 


2.72 


3.89 


5.34 


7.11 



565 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 



Point. 


f-H 


5. 


0050305l>-^0005eOO 


ir>c»-<;j<05-<;t<a>coc>.r-(^ 

r-H .— 1 1— 1 r-( (N CO 


1 


l-^IO OOIOOICSI 05lt^ t- C^ CO i 
t-t-OOCOCOt^OCCOOrH 
C^ CD CO Tfl C^ in CQ »;^ t>- CO 
CO CO OO r-H C>. 05 CO -^ CO .-< 
O C^ r-H lO !>. C35 r— 1 CO ■^ CD 


C»050000.-lr-l.-l,-l 


» 


1 

cr> in t^ n-1 c<j CO 00 OS CN t- 1 
o '^cq c^ojoo Oi in r-H ' 
r-H m in (N 05 05 CN o c>- CO 


oOrHcoinoiincsjOrH ; 


-e- 

> 


i 

CD 00 t>. OilCN 05 CMI05 OICO 
CDOOOOCO(NOOOOCDinO 

o^cDoc<icocsia).-Hco 
oococooococsjc-coooin 
in in --H in 05 <M Tj< CD 00 o j 


in CD C>- t- O 00 OO 00 00 CD 

i 


o 

ho 

o 
—J 


1 

ICO inioi-"^ o-icDiotoi^ CO 

Cr> 00 Ttl CO eg CD CD CM CO 05 
OSOiOiOOCDCNCDOOinO- 
CJ5 CD CD 05 C35 CJ> OO C^ CO -^l) 
CD 05 CT> 05 05 05 05 03 05 Oi . 


Oi Oi Oi OO Oi 0> Oi Oi 03 Oi i 


-e- 

bO 
o 


•«# (N OI05 OlCOIt^ 00 rH\<Z> 
00a>00CSlC^COCDO500T*< 

oc-oooinocooxN'^ 

^^^T^^CDOOOOOOCD 
a>r^C>.03>— ICNCOTiHinCD 


t<-000000C»CT>O3CDCDO> 


o 


IC5ICDICO <NJlTj)|(N COlO 03 O 
050>00CDrHC0Oi— ICOC^ 
CT>O5CJJa>C3500C»-inCM00 
O5C35CDCDCDCDCDCDCD00 






IC^ CSJICOIrHliniCNJ Oi O t^ic> 

oocoogi>.OCQ<— lOJCqi-H 
oc>qinoocoooT*<oooco 

OOOOr-ln-lCNCOCO"«;*< 




Total 
Central Angle 


oooooooooo 

CO CO O O CO CO O O CO CO 

OrHco»nt>0'«*oo(Nc>* 

l-HrHrHCqCS 


"c2 


i-HCsico-«^in«ot>oooO 
1— 1 



t^CDin-^CNrHOSC"- 



•^COCSJrHOSOOCD^CN 



I— ioa>ooc»-mco<N 



03 00 t>. CD •^ CO I— I 



eg O c^ IT) eg o 
Tt< o o o in CO 



in T*< CO i-t 



I— i CO in t^ 



o o o o o 

O CO O CO o 



in t«- o eg in 
•^ o eg eg 



Tf ■■^ CO eg 



1— I CO ^ CD 00 





o 

CO 


o 
o 


o 

CO 


o 
o 


§ 


o 

o 


o 

CO 


o 

o 


o 

CO 


in 00 

m 1— 1 


^ 


o 


in eg o 
CO in o 


o 
o 


in 

I— I 


CO 


o 
I— < 


eg 
in 






o 

CO 


eg r-l rH 


o 


r-H 


eg 


■^ 


in 


t«- Oi 




o 
o 


o o 
CO o 


o 
o 


o 
o 


o 

CO 


o 
o 


o 

CO 


CD 

in 


rH O 

eg Tt< 


w 


o 
in 


eg in 


§ 


s 


o 


in 
eg 


eq 

in 


C35 

eg 


rH r-i 




rH 


t-H O 


o 


-■ 


CO 


CO 


•^ 


CD 


00 o 




o 

CO 


§ 


o 

o 


o o 
o CO 


o 
o 


o 

CO 


00 

in 


eg 


in o 

T}1 o 


« 


eg 
in 


o 

CO 


o 
o 


in t* 

-<*< CO 


o 


eg 
in 


^ 




05 eg 
eg eg 




•^o 


o 


o 


o 


i-H 


eg 


CO 


in 


CD 


00 o 



Oi-Hi— tcgeomcoooo 



o o o o o 

O CO O CO o 



O 



eg CO ^ CD t>- oa 



•-.5 I 



^ y-i 



566 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 
Dimensions of various 2°-per-25-feet spirals. — Part B. 



Deg. 
of 


L'gth 
of 


ZK 


QK 


A'N 


NQ 


A' to Z. 


^of- 








curve. 


spiral 










Defl. 


Dist. 


curve. 


4'' 00' 


25 


0.11 


25.00 


0.05 


12.50 


0° 


15' 


12.50 


4° 00' 


20 


50 


0.55 


50.00 


0.09 


15.38 





45 


34.62 


20 


40 


50 


0.55 


50.00 


0.12 


17.85 





45 


32.14 


40 


5° 00' 


50 


0.55 


50.00 


0.15 


19.99 





45 


30.00 


5° 00' 


20 


50 


0.55 


50.00 


0.18 


21,86 





45 


28.13 


20 


40 


50 


0.55 


50.00 


0.20 


23.52 





45 


26.47 


40 


6^ 00' 


50 


55 


50 00 


0.22 


24 99 


0° 


45' 


25.00 


6° 00' 


20 


75 


1.52 


74.98 


0.29 


27.61 




30 


47.37 


20 


40 


75 


1.52 


74 98 


0.35 


29.97 




30 


45.00 


40 


7° 00' 


75 


1.52 


74.98 


0.40 


32.11 




30 


42.86 


7° 00' 


20 


75 


1.52 


74.98 


0.46 


34.06 




30 


40.91 


20 


40 


75 


1.52 


74.98 


0.50 


35.84 




30 


39.13 


40 


8^ 00' 


75 


1.52 


74.98 


0.54 


37.46 


1° 


30' 


37.50 


8° 00' 


20 


100 


3.27 


99.92 


0.65 


39.94 


2 


30 


60.00 


20 


40 


100 


3.27 


99.92 


0.75 


42 . 24 


2 


30 


57.69 


40 


9° 00' 


100 


3.27 


99.92 


0.85 


44.37 


2 


30 


55.56 


9° 00' 


20 


100 


3.27 


99.92 


0.93 


46.35 


2 


30 


53.57 


20 


40 


100 


3.27 


99.92 


1.01 


48.20 


2 


30 


51.72 


40 


lO'' 00' 


100 


3.27 


99 92 


1.09 


49.92 


2° 


30' 


50.00 


10° 00' 


20 


125 


5.99 


124 77 


1,24 


52.30 


3 


45 


72.58 


20 


40 


125 


5.99 


124.77 


1,39 


54,55 


3 


45 


70.31 


40 


11° 00' 


125 


5.99 


124.77 


1,53 


56,68 


3 


45 


68.18 


11° 00' 


20 


125 


5.99 


124.77 


1.66 


58.67 


3 


45 


66.18 


20 


40 


125 


5.99 


124.77 


1.78 


60.55 


3 


45 


64.29 


40 


12° 00' 


125 


5.99 


124,77 


1.90 


62.33 


3° 


45' 


62.50 


12° 00' 


20 


150 


9 90 


149,46 


2.11 


64.68 


5 


15 


85.14 


20 


40 


150 


9 90 


149,46 


2.31 


66 86 


5 


15 


82.89 


40 


13° 00' 


150 


9,90 


149 46 


2.51 


68 97 


5 


15 


80.77 


13° 00' 


20 


150 


9 90 


149 46 


2.69 


70.97 


5 


15 


78.75 


20 


40 


150 


9.90 


149.46 


2.87 


72.88 


5 


15 


76.83 


40 


14° 00' 


150 


<9 90 


149.46 


3 03 


74.69 


5* 


15' 


75.00 


14° 00' 


20 


175 


15.21 


173 89 


3 30 


76.93 


7 


00 


97.67 


20 


40 


175 


15.21 


173.89 


3,57 


79.12 


7 


00 


95.45 


40 


15° 00' 


175 


15-21 


173.89 


3 83 


81.22 


7 


00 


93.33 


15° 00' 


20 


175 


15.21 


173,89 


4,08 


83.22 


7 


00 


91.30 


20 


40 


175 


15.21 


173.89 


4.31 


85.14 


7 


00 


89.36 


40 


16° 00' 


175 


15.21 


173.89 


4.54 


86.98 


7° 


00' 


87.50 


16° 00' 


20 


200 


22.10 


197.92 


4.87 


89.15 


9 


00 110.20 


20 


40 


200 


22.10 


197.92 


5.21 


91.31 


9 


00 108.00 


40 


17° 00' 


200 


22.10 


197.92 


5.54 


93.39 


9 


00 105.88 


17° 00' 


20 


200 


22.10 


197.92 


5.86 


95.38 


9 


00 103.85 


20 


40 


200 


22.10 


197.92 


6.16 


97.30 


9 


00 


101.89 


40 


18° 00' 


200 


22.10 


197.92 


6.45 


99.15 


9° 


00' 


100.00 


18° 00' 


20 


225 


30.75 


221.38 


6 86 


101.27 


11 


15 


122.73 


20 


40 


225 


30.75 


221.38 


7.28 


103.39 


11 


15 


120.54 


40 


19° 00' 


225 


30.75 


221.38 


7 69 


105.45 


11 


15 


118.42 


19° 00' 


20 


225 


30.75 


221.38 


8.09 


107.43 


11 


15 


116.38 


20 


40 


225 


30.75 


221.38 


8.47 


109.34 


11 


15 


114.41 


40 


20° 00' 


225 


30.75 


221.38 


8 83 


111.19 


11° 


15' 


112.50 


20° 00' 


20 


250 


41.32 


244.03 


9.31 


113.23 


13 


45 


135.25 


20 


40 


250 


41.32 


244.03 


9.82 


115.32 


13 


45 


133.06 


40 


21° 00' 


250 


41.32 


244.03 


10.32 


117.35 


13 


45 


130.95 


21° 00' 


20 


250 


41.32 


244.03 


10.80 


119.30 


13 


45 


128.91 


20 


40 


250 


41.32 


244.03 


11.26 


121.20 


13 


45 


126.92 


40 


22° 00' 


250 


41.32 


244.03 


11.71 


123.04 


13° 


45' 


125.00 


22° 00' 



567 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 

Dimensions of various 2°-per-25-feet spirals. — Part C. 

Values of AX. 





Dej^ree 


of curve. 






A 


























1 


1 






4° 


6° 


8° 


10° 


12° 


14° 


16° 


18° 1 


20° 1 


22° 


4°i 


0.00 


0.01 


0.02 


0.04 0.07 


0.11 


0.18 


0.23 


0.31 0.41 


6 


00 


0.01 


0.03 


0.06 0.10 


0.16 


0.24 


0.34 


0.46 0.61 


8 


0.00 


0.02 


0.04 


0.08 0.13 


21 


0.32 


0.45 


0.62 


0.82 


10 


0.00 


0.02 


0.05 


0.10 0.17 


0.27 


0.40 


0.56 


0.77 


1.02 


12 


0.01 


0.02 


0.06 


0.11 0.20 


32 


0.48 


0.68 


0.93 


1.23 


14 


0.01 


0.03 0.07 


0.13 i 0.23 


0.37 


0.56 


0.79 1 


1.08 


1.44 


16 


0.01 


0.03 0.08 


0.15 27 


0.43 


0.64 


0.91 1 


1.24 


1.65 


18 


0.01 


0.03 0.09 


0.17 0.30 


0.48 


0.72 


1.02 


1.40 


1.85 


20 


0.01 


0.04 0.10 


0.19 1 


0.34 


0.53 


0.80 


1.14 


1.56 


2.06 


22 


0.01 


0.04 0.11 


0.2li 


0.37 


0.59 


0.88 


1.25 


1.72 


2.28 


24 


0.01 


0.05 


0.12 


0.23 


40 


0.64 


0.96 


1.37 


1.88 


2.49 


26 


0.01 


0.05 


0.13 


0.25 


0.44 


0.70 


1.05 


1.49 


2.04 


2.70 


28 


0.01 


0.05 


0.14 


0.27 


0.47 


0.76 


1.13 


1.61 


2.20 


2.92 


30 


0.01 


0.06 


0.15 


0.29 


0.51 


0.81 


1.22 


1.73 


2.37 


3.14 


32 


0.02 


0.06 


0.16 


0.31 


0.54 


0.87 


1.30 


1.85 


2.53 


3.36 


34 


0.02 


07 


0.17 


33 58 


93 


1.39 


1.97 


2.70 


3 58 


36 


0.02 


0.07 


0.18 


0.35 


62 


0.99 


1.47 


2.10 


2 87 


3 80 


38 


0.02 


0.08 


0.19 


37 


0.65 


1.04 


1.56 


2.22 


3.04 


4.03 


40 


0.02 


0.08 


0.20 


0.40 


0.69 


1.10 


1.65 


2.35 


3.22 


4.26 


42 


0.02 


0.08 


0.21 


0.42 


0.73 


1.16 


1.74 


2.48 


3.39 


4.49 


44 


0.02 


0.09 


0.22 


0.44 


77 


1.23 


1.83 


2.61 


3.57 


4.73 


46 


0.02 


09 


0.23 


0.46 


0.81 


1.29 


1.93 


2.74 


3.75 


4.97 


48 


0.02 


0.10 


0.24 


0.48 


0.85 


1.35 


2.02 


2.87 


3.93 


5.21 


50 


0.03 


0.10 


0.25 


0.51 


0.89 


1.41 


2.11 


3.01 


4.12 


5.46 


52 


0.03 


0.11 


0.27 


0.53 


93 


1.48 


2.21 


3.15 


4.31 


5.71 


54 


0.03 


0.11 


0.28 


0.55 


97 


1.54 


2.31 


3.29 


4.50 


5.97 


56 


0.03 


0.12 


29 


0.58 


1,01 


1.61 


2.41 


3.43 


4.70 


6.23 


58 


0.03 


0.12 


0.30 


0.60 


1.05 


1.68 


2.51 


3.58 


4.90 


6.49 


60 


0.03 


0.13 


0.31 


0.63 


1.10 


1.75 


2.62 


3.73 


5.10 


6.76 


62 


0.03 


0.13 


0.33 


0.65 


1.14 


1.82 


2.73 


3.88 


5.31 


7.04 


64 


0.03 


0.14 


0.34 


0.68 


1.19 


1.90 


2.83 


4.03 


5.52 


7.32 


66 


0.03 


0.14 


0.35 


0.71 


1 23 


1.97 


2.95 


4.19 


5.74 


; 7.60 


68 


0.04 


0.15 


0.37 


0.73 


1.28 


2.05 


3.06 


4.35 


5.96 


7.90 


70 


0.04 


0.15 


0.38 


0.76 


1.33 


2.12 


3.18 


4.52 


6.19 


8.20 


72 


0.04 


0.16 


0.40 


0.78 


1.38 


2.20 


3 30 


4 69 


6.42 


8.51 


74 


0.04 


0.16 


0.41 


0.81 


1.43 


2.28 


3.42 


4 86 


6.66 


8 82 


76 


0.04 


0.17 


0.43 


0.84 


1.48 


2.37 


3.54 


5.04 


6.90 


9.15 


78 


0.04 


0.18 


0.44 


0.88 


1.54 


2.46 


3.67 


5.22 


7.15 


9 48 


80 


0.05 


0.18 


0.46 


0.91 


1.59 


2.54 


3.81 


5.41 


7.41 


9.82 


82 


0.05 


0.19 


0.47 


0.94 


1.65 


2.64 


3.94 


5.61 


7.68 


10.18 


84 


0.05 


0.20 


0.49 


0.98 


1.71 


2.73 


4.08 


5.81 


7.95 


10.54 


86 


0.05 


0.20 


0.51 


1.02 


1.77 


2 83 


4.23 


6.02 


8.24 


10.92 


88 


0.05 


0.21 


0.53 


1.05 


1.83 


2.93 


4.38 


6.23 


8.53 


11.31 


90 


0.05 


0.22 


0.55 


1.09 


1.90 


3.03 


4.54 


6.45 


8.83 


11.71 


92 


0.06 


0.23 


0.56 


1.13 


1.97 


3.14 


4.70 


6.68 


9.15 


12.12 


94 


0.06 


0.23 


0.58 


1.17 


2.04 


3.25 


4.86 


6.92 


9.47 


12.56 


96 


0.06 


0.24 


0.61 


1.21 


2.11 


3.37 


5.04 


7.17 


9.81 


13.00 


98 


! 0.06 


0.25 


0.63 


1.25 


2.19 


3.49 


5.22 


7.42 


10.16 


13.47 


100 


06 


0.26 


0.65 


1.30 


2.26 


3.61 


5.41 


7.69 




10.53 


13.95 



568 



TABLE V.> -LOGARITHMS OF NUMBFLiy. 



N. 





1 


2 


3 

130 


4 

173 


5 

216 


6 

260 


7 
303 


8 

346 


9 


P.P. 


100 


GO 000 


043 


087 


389 


m7=: m ^ . ^ . ^ 






















4c> 4iJ q:'a 41 


iUi 


432 


475 


518 


561 


604 


646 


689 


732 


775 


817 


•1 


4.3 


4.3 


4.2 


4.1 


i02 


860 


902i 945 


987 


*030 


*072 


*114 


*157 


*199 


*241 


• 2 


8 


7 


8 


• fi 


8 


.4 


8 


2 


103 


01 283 


326! 368 


410 


452 


494 


536 


578 


619 


661 


• 3 


13 


.0 


12 


■ 9 


12 


• R 


12 


3 


104 


703 


745! 787 


828 


870 


911 


953 


994 


*036 


*077 


•4 


17 


.4 


17 


.2 


16 


8 


16 


.4 


105 


02 119 


160' 201 


243 


284 


325 


366 


407 


448 


489 


5 


21 


• 7 


21 


.5 


21 


.0 


20 


5 


106 


530 


571 612 


653 


694 


735 


775 


816 


857 


898 


• 6 


26 


•1 


25 


8 


25 


• 2 


24 


6 


107 


938 


979i*019 


*060 


*100 


*141 


*181 


*221 


*262 


*302 


•7 


30 


• 4 


30 


•] 


29 


4 


28 


.7 


108 


03 342 


382! 422 


463 


503 


543 


583 


623 


663 


703 


8 


34 


8 


34 


.4 


33 


• 6 


32 


8 


109 


742 
04 139 


782 822i 862 


901 
297 


941 
336 


981 


*020 
415 


*060 
454 


*100 
493 


9 


39 


•1 


38 


•7 


37 


8 


36 


9 


110 


178 


218 


257 


375 


mTZ m^ ^^ »» 


111 


532 


571 


610 


649 


688 


727 


766 


805 


844 


883 


•1 


4.0 


4U 

4.0 


39 


88 


112 


922 


960 


999 


*038 


*076 


^115 


*154 


*192 


*231 


*269 


• 2 


8 


•1 


8 





7 


8 


7 


6 


113 


05 308 


346| 384 


423 


461 


499 


538 


576 


614 


652 


3 


12 


.1 


12 


.0 


n 


• 7 


11 


4 


114 


690 


728: 766 


804 


842 


880 


918 


956 


994 


*032 


■ 4 


16 


.2 


16 


.0 


15 


.6 


15 


.2 


115 


06 070 


1071 145 


183 


220 


258 


296 


333 


371 


408 


• 5 


20 


.2 


20 





19 


5 


19 





116 


446 


483 520 


558 


595 


632 


670 


707 


744 


78] 


• 6 


24 


• 3 


24 


.0 


23 


4 


22 


8 


117 


818 


855 893 


930 


967 


*004 


*040 


*077 


*114 


*151 


• 7 


28 


. 3 


28 


.0 


27 


•3 


26 


6 


118 


07 188 


225 261 


298 


335 


372 


408 


445 


481 


518 


8 


32 


./ 


32 


.0 


31 


2 


30 


4 


119 


554 


591 


627 


664 


700 


737 


773 


809 


845 


882 


.9 


36 


.1 


36 





35 


1 


34 


2 


120 


918 


954 


990 


*026 


*062 


*098 


*184 


*170 


*206 


*242 























— • — I 




37 37 36 35 


121 


08 278 


314 


350 


386 


422 


457 


493 


529! 564 


600 


.1 


3.7 


3.7 


36 


35 


122 


636 


671 


707 


742 


778 


813 


849 


884 920 


955 


-2 


7 


5 


7 


4 


7 


2 


70 


123 


990 


*026 


*061 


*096 


*131 


*166 


*202 


*237*272 


*307 


• 3 


11 


2 


11 


1 


10 


8 


10.5 


124 


09 342 


377 


412 


447 


482 


517, 552; 586 621 


656 


• 4 


15 





14 


8 


14 


4 


14.0 


125 


691 


725 


760 


795 


830 


864) 899i 933 968 


*002 


5 


18 


7 


18 


5 


18 





17.5 


126 


10 037 


071 


106 


140 


174 


209 


243 277i 312 


346 


■ 6 


22 


5 


22 


2 


21 


6 


21.0 


127 


380 


414 


448 


483 


517 


551 


585' 619 653 


687 


7 


26 


2 


25 


9 


25 


2 


24.5 


128 


721 


755 


789 


822 


856 


890 


924' 958| 991 


*025 


8 


30 





29 


6 


28 


8 


28.0 


129 


11 059 


092 


126 


160 


193 


227 


260! 294 327 


361 


• 9 


33. 7 


33-3 


32 


4 


31.5 


130 


394 


427 


461 


494 


528 


561 


594 


627 


661 


694 


35 34 33 32 


131 


727 


760 


793 


826 


859 


892 


925 


958 


991 


*024 


• 1 


3 4 


3 4 


33 


32 


132 


12 057 


090 


123 


156 


189 


221 


254 


287 


320 


352 


• 2 


6 


g 


6 


8 


6 


6 


6. 


4 


133 


385 


418 450 


483 


515 


548 


580 


613 


645 


678 


3 


10 


3 


10 


2 


9 


9 


9. 


6 


134 


710 


743! 775 


807 


840 


872 


904 


9371 969 


*001 


.4 


13 


8 


13 


6 


13 


2 


12. 


8 


135 


13 033 


065 097 


130 


162 


194 


226 


258; 290 


322 


• 5 


17 


2 


17 





16 


5 


16. 





136 


354 


386' 417 


449 


481 


513 


545 577 608 


640 


• 6 


20 


7 


20 


4 


19 


8 


19. 


2 


137 


672 


703 735 


767 


798 


830 


862 893 925 


956 


•7 


24 


1 


23 


8 


23 


1 


22. 


4 


138 


988 


*019 *051 


*082 


*n8 


*145 *176 *207*239 


*270 


8 


27 


6 


27 


2 


26 


4 


25. 


6 


139 


14 301 


332 


364 


395 


426 


457j 488! 519 550 


582 


• 9 


31. 





30. 


6 


29.7 


28.8 


140 


613 


644 


675 


706 


736 


7671 798j 829! 860 


891 


31 31 30 29 


141 


922 


952 


983 


*014 


*045 


*075 *106*137*167 


*198 


• 1 


3.1! 


3.1 


30 


2.9 


142 


15 229 


259 290 


320 


351 


38li 412' 442, 473 


503 


.2 


6. 


3 


6 


2 


6. 





5.8 


143 


533 


564 


594 


624 


655 


685! 715! 745! 776 


806 


3 


9 


4 


9 


3 


9 





8.7 


144 


836 


866 


896 


926 


956 


987*017*047*077 


*107 


• 4 


12 


6 


12 


4 


12 





11.6 


145 


16 137 


166 


196 


226 


256 


286! 3161 346: 376 


405 


• 5 


15 


7 


15 


5 


15 





14.5 


146 


435 


465 


4941 524 


554 


584 


613 


643 672 


702 


.6 


18 


9 


18 


6 


18 





17.4 


147 


731 


761 


791 


820 


849 


879 


908 


938 


967 


997 


• 7 


22 





21 


7 


21 





20.3 


148 


17 026 


055 


085 


114 


143 


172 


202 


231 


260 


289 


8 


25 


2 


24 


8 


24. 





23.2 


149 


318 


348 


377 


406 


435 


464 


493 


522 


551 


580 


• 9 


28 


3 


27 


9 


27 





26.1 


150 


609 


638 


667 
2 


696 
3 


725 
4 


753 
5 


782 
6 


811 

7 


840 


869 




n 


N. 





1 


8 


9 








P. 


r 


• 







569 









TABLE \ 


.—LOGARITHMS 


OF NUMBERS 














N. 





1 


2 


3 


4 


5 


6 


7 

811 


8 
840 


9 

869 


P. P. 


150 


17 609 


638 


667 


696 


725 


753 


782 
































29 28 27 


151 


897 


926 


955 


984 


^012 


^041 


*070 


*098 


=^127 


^156 


.1 


2-9 


28 


27 


152 


18 184 


213 


241 


270 


298 


327 


355 


384 


412 


440 


• 2 


5 


8 


5 


• 6 


5 


4 


153 


469 


497 


526 


554 


582 


611 


639 


667 


695 


724 


3 


8 


•7 


8 


•4 


8 


•1 


154 


752 


780 


808 


836 


864 


893 


921 


949 


977 


=^005 


• 4 


11 


• 6 


11 


.2 


10 


8 


155 


19 033 


061 


089 


117 


145 


173 


201 


229 


256 


284 


• 5 


14 


5 


14 





13 


• 5 


156 


312 


340 


368 


396 


423 


451 


479 


507 


534 


562 


.6 


17 


■ 4 


16 


8 


16 


■2 


157 


590 


617 


645 


673 


700 


728 


755 


783 


810 


838 


•7 


20 


• 3 


19 


6 


18 


9 


158 


865 


893 


920 


948 


975 


^^003 


*030 


*057 


^085 


'ni2 


.8 


23 


• 2 


22 


4 


21 


6 


159 


20 139 


167 


194 


221 


249 


276 


303 


330 


357 


385 


.9 


26 


• 1 


25 


.2 


24 


• 3 


160 


412 


439 


466 


493 


520 


547 


574 


601 


628 


655 


26 26 
























161 


682 


709 


736 


763 


790 


817 


844 


871 


898 


924 


•11 


2-61 


2-6 


162 


951 


978 


=^=005 


*032 


*058 


^085 


*112 


*139 


n65 


^192 




• 2 


5 


• 3 


5 


.2 


163 


21 219 


245 


272 


298 


325 


352 


378 


405 


431 


458 




• 3 


7 


•9 


7 


8 


164 


484 


511 


537 


564 


590 


616 


643 


669 


695 


722 




• 4 


10 


6 


10 


4 


165 


748 


774 


801 


827 


853 


880 


906 


932 


958 


984 




• 5 


13 


• 2 


13 





166 


22 01 . 


037 


063 


089 


115 


141 


167 


193 


219 


245 




• 6 


15 


■ 9 


15 


6 


167 


27. 


297 


323 


349 


375 


401 


427 


453 


479 


505 




• 7 


18 


5 


18 


.2 


168 


531 


557 


582 


608 


634 


660 


686 


711 


737 


763 




8 


21 


• 2 


20 


8 


169 


788 


814 


840 


865 


891 


917 


942 


968 


994 


=^019 




.9 


23 


■ 8 


23-4 


170 


23 045 


070 


096 


121 


147 


172 


198 


223 


249 


274 




171 


299 


325 


350 


375 


401 


426 


45l 


477 


502 


527 


• 1 


/CO 

2-5 


2-5 


2.4 


172 


553 


578 


603 


628 


653 


679 


704 


729 


754 


779 


• 2 


5 


•1 


5 


• 


48 


173 


804 


829 


855 


880 


905 


930 


955 


980 


*005 


*030 


3 


7 


• 6 


7 


• 5 


7-2 


174 


24 055 


080 


105 


129 


154 


179 


204 


229 


254 


279 


• 4 


10 


■ 2 


10 


• 


9.6 


175 


304 


328 


353 


378 


403 


427 


452 


477 


502 


526 


• 5 


12 


■7 


12 


■ 5 


12.0 


176 


551 


576 


600 


625 


650 


674 


699 723 


748 


773 


• 6 


15 


• 3 


15 


.0 


14.4 


177 


797 


822 


846 


871 


895 


920 


944 968 


993 


*017 


• 7 


17 


• 8 


17 


.5 


16.8 


178 


25 042 


066 


091 


115 


139 


164 


188 


212 


237 


261 


.8 


20 


■4 


20 


.0 


19-2 


179 


285 


309 


334 


358 


382 


406 


430 


455 


479 


503 


-9 


22.9 


22.5 


21.6 


180 


527 


55l 


575 


599 


623 


647 


672 


696 


720 


744 


23 23 


181 


768 


792 


816 


840 


863 


887 


911 


935 


959 


983 


•11 


2.31 


2-3 


182 


26 007 


031 


055 


078 


102 


126 


150 


174 


197 


221 




2 


4 


•7 


4 


6 


183 


245 


269 


292 


316 


340 


363 


387 


411 


434 


458 




3 


7 





6 


9 


184 


482 


505 


529 


552 


576 


599 


623 


646 


670 


693 




4 


9 


4 


9 


2 


185 


717 


740 


764 


787 


811 


834 


858 


881 


904 


928 




5 


11 


7 


11 


5 


186 


951 


974 


998 


*021 


*044 


*068 


=^091 


^114 


n37 


*161 




6 


14 


1 


13 


8 


187 


27 184 


207 


230 


254 


277 


300 


323 


346 


369 


392 




7 


16 


4 


16 


1 


188 


416 


439 


462 


485 


508 


531 


554 


577 


600 


623 




8 


18 


8 


18 


4 


189 


646 


669 


692 


715 


738 


761 


784 


806 


829 


852 


.9! 


21-11 


20.7 


190 


875 


898 


921 


944 


966 


989 


*012 


*035 


*058 


*080 


22 22 21 


191 


28 103 


126 


149 


171 


194 


217 


239 


262 


285 


307 


.1 


2.2 


2.2 


2.1 


192 


330 


352 


375 


398 


420 


443 


465 


488 


510 


533 


.2 


4 


• 5 


4 


•4 


4 


3 


193 


555 


578 


600 


623 


645 


668 


690 


713 


735 


758 


• 3 


6 


•7 


6 


6 


6 


4 


194 


780 


802 


825 


847 


869 


892 


914 


936 


959 


981 


• 4 


9 


.0 


8 


8 


8 


6 


195 


29 003 


025 


048 


070 


092 


114 


137 


159 


181 


203 


• 5 


11 


.2 


11 





10 


7 


196 


225 


248 


270 


292 


314 


336 


358 


380 


402 


424 


• 6 


13 


. 5 


13 


.2 


12 


9 


197 


446 


468 


490 


512 


534 


556 


578 


600 


622 


644 


• 7 


15 


. 7 


15 


.4 


15 





198 


666 


688 


710 


732 


754 


776 


798 


820 


841 


863 


.8 


18 


.0 


17 


.6 


17 


2 


199 


885 


907 


929 


950 


972 


994 


*016 


*038 


=^059 


*081 


.9 


20.2 


19.819.3 


200 


30 103 


124 


146 


168 


190 
4 


211 
5 


233 
6 


254 

7 


276 
8 


298 
9 




N. 





1 


2 


3 






P 


. F 









570 









TABLE v.— LOGARITHMS ( 


OF NUMBERS. 






X. 





1 


2 

146 


3 

168 


4 

190 


5 

211 


6 

233 


7 
254 


8 
276 


9 


P. P. 


JJOO 


30 103 


124 


298 


























22 31 


201 


319 


341 


363 


384 


406 


427 


449 


470 


492 


513 


.1 2.2 


2.1 


202 


535 


556 


578 


599 


621 


642 


664 


685 


707 


728 


.2 4.4 


4. 


2 


203 


749 


771 


792 


813 


835 


856 


878 


899 


920 


941 


.3 6-6 


6. 


3 


204 


963 


984 


*005 


*027 


=^=048 


^069 


*090 


^112 


*133 


^154 


• 4 88 


8. 


4 


205 


31 175 


196 


217 


239 


260 


281 


302 


323 


344 


365 


.5 11-0 


10. 


5 


206 


386 


408 


429 


450 


471 


492 


513 


534 


555 


576 


.6 13-2 


12. 


6 


207 


597 


618 


639 


660 


681 


702 


722 


743 


764 


785 


.7 15-4 


14. 


7 


208 


806 


827 


848 


869 


890 


910 


931 


952 


973 


994 


.8 17.6 


16. 


8 


209 


32 014 


035 


056 


077 


097 


118 


139 


160 


180 


201 


.9 19. 8 


18. 


9 


310 


222 


242 


263 


284 


304 


325 


346 


366 


387 


407 

















— 


— — 










20 20 


^11 


428 


449 


469 


490 


510 


531 


551 


572 


592 


613 


.1 2.0 


2.0 


212 


633 


654 


674 


695 


715 


736 


756 


776 


797 


817 


.2 4.1 


4 





213 


838 


858 


878 


899 


919 


94G 


960 


980 


=*^001 


*021 


.3 6.1 


6 





214 


33 04l 


061 


082 


102 


122 


142 


163 


183 


203 


223 


.4 8.2 


8 





215 


244 


264 


284 


304 


324 


344 


365 


385 


405 


425 


.5 10.2 


10 





216 


445 


465 


485 


505 


525 


546 


566 


586 


606 


626 


• 6 12. c 


12 





217 


646 


666 


686 


706 


726 


746 


766 


786 


806 


825 


.7 14.3 


14 





218 


845 


865 


885 


905 


925 


945 


965 


985 


*0C4 


*024 


• 8 16.4 


16 





219 


34 044 


064 


084 


104 


123 


143 


163 


183 


203 


222 


.9 IS. 4 


18 





230 


242 


262 


28l 


301 


321 


341 


360 


380 


400 


419 
































19 19 


221 


439 


459 


478 


498 


518 


537 


557 


576 


596 


615 


.1 l.c 


1.9 


222 


635 


655 


674 


694 


713 


733 


752 


772 


791 


811 


.2 Si 


3 


8 


223 


830 


850 


869 


889 


908 


928 


947 


966 


986 


'*=G05 


.3 5. J 


\ 5 


7 


224 


35 025 


044 


063 


083 


102 


121 


141 


160 


179 


199 


.4 71 


1 7 


6 


225 


218 


237 


257 


276 


295 


314 


334 


353 


372 


391 


.5 9.^ 


r 9 


5 


226 


411 


430 


449 


468 


487 


507 


526 


545 


564 


583 


.6 11.' 


J 11 


4 


227 


602 


621 


641 


660 


679 


698 


717 


736 


755 


774 


.7 13. f 


5 13 


3 


228 


793 


812 


831 


850 


869 


888 


907 


926 


945 


964 


.8 15. f 


515 


.2 


229 


983 
36 173 


*002 
191 


*021 
210 


*040 


*059 
248 


=*^078 


*C97 


=*=116 


*135 


*154 


•9 17. f 


)17 


1 


330 


229 


267 


286 


305 


323 


342 


























18_ 18 


231 


361 


380 


399 


417 


436 


455 


47-^ 


492 


511 


530 


.11 l.i 


! 1.8 


232 


549 


567 


586 


605 


623 


642 


661 


679 


698 


717 


.2 3.' 


7 3 


•6 


233 


735 


754 


773 


791 


810 


828 


847 


866 


884 


903 


.3 5.f 


5 5 


• 4 


234 


921 


940 


958 


977 


996 


^014 


^C3S 


^O-il 


*070 


*088 


.4 7.^ 


I 7 


.2 


235 


37 107 


125 


143 


162 


180 


199 


217 


236 


254 


273 


.5 9.1 


I 9 





236 


291 


309 


328 


346 


364 


383 


401 


420 


438 


456 


.6 11.: 


10 


8 


237 


475 


493 


511 


530 


548 


566 


584 


603 


621 


639 


.7 12. < 


) 12 


.6 


238 


657 


676 


694 


712 


730 


749 


767 


785 


803 


821 


.8 14.1 


1 14 


.4 


239 


840 


858 


876 


894 


912 


930 


948 


967 


985 


=^T03 


.9 16. ( 


) 16 


2 


240 


38 021 


039 


057 


075 


093 


111 


129 


147 


165 


183 
































17 17 


241 


201 


219 


237 


255 


273 


291 


3uy 


32V 


345 


363 


• 1 !.■; 


^ 1.7 


242 


381 


399 


417 


435 


453 


471 


489 


507 


525 


543 


.2 3. J 


) 3 


4 


243 


560 


578 


596 


614 


632 


650 


667 


685 


703 


721 


.3 5.^ 


5 


1 


244 


739 


757 


774 


792 


810 


828 


845 


863 


881 


899 


.4 7.C 


6 


8 


245 


916 


934 


952 


970 


987 


*005 


*023 


=*=040 


*058 


*076 


.5 8.^ 


8 


5 


246 


39 093 


111 


129 


146 


164 


181 


199 


217 


234 


252 


.6 10. e 


10 


2 


247 


269 


287 


305 


322 


340 


357 


375 


392 


410 


427 


.712.2 


11 


9 


248 


445 


462 


480 


497 


515 


532 


550 


567 


585 


602 


• 8 14. C 


13 


6 


249 


620 


637 


655 


672 


689 


707 


724 


742 


759 


776 


.9115.7 


15 


3 


250 


794 


811 


828 
2 


846 
3 


863 
4 


881 
5 


898 
6 


915 

7 
1 


933 


950 
9 




N. 





1 


8 


P. 


P. 





571 









TABLE V 


.—LOGARITHMS OF NUMBERS. 










N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


350 


39 794 


811 


828 


846 


863 


881 


898 


915 


933 


950 




251 


967 


984 


*002 


*019 


*036 


*054 


*071 


*088 


*105 


*123 




252 


40 140 


i 157 


174 


191 


209 


226 


243 


260 


277 


295 


17 17 


253 


312 


329 


346 


363 


380 


398 


415 


432 


449 


466 


• 1 


1.7 


1.7 


254 


483 


500 


517 


534 


551 


569 


586 


603 


620 


637 


• 2 


3 


5 


3 


4 


255 


654 


671 


688 


705 


722 


739 


756 


773 


790 


807 


• 3 


5 


2 


5 


1 


256 


824 


841 


858 


875 


892 


908 


925 


942 


959 


976 


• 4 


7 





6 


8 


257 


993 


*010 


*027 


*044 


*061 


*077 


-^094 


*111 


*128 


*145 


• 5 


8 


7 


8 


5 


258 


41 162 


179 


195 


212 


229 


246 


263 


279 


296 


313 


• 6 


10 




10 


2 


259 


330 


346 


363 


380 


397 


413 


430 


447 


464 


480 


• 7 
.8 

• 9 


12 
14 
15 


? 


11 
13 
15 


9 
6 
3 


260 


497 


514 


530 


547 


564 


581 


_597 


614 


631 


647 


261 


664 


680 


697 


714 


730 


747 


764 


780 


797 


813 




262 


830 


846 


863 


880 


896 


913 


929 


946 


962 


979 




263 


995 


*012 


*028 


*045 


*061 


*078 


=^094 


*111 


*127 


*144 




264 


42 160 


177 


193 


209 


226 


242 


259 


275 


292 


308 




265 


324 


341 


357 


373 


390 


406 


423 


439 


455 


472 


16 16 


266 


488 


504 


521 


537 


553 


569 


586 


602 


618 


635 


.1 


1.6 


1-6 


267 


651 


667 


683 


700 


716 


732 


748 


765 


781 


797 


• 2 


3 


3 


3 


2 


268 


813 


829 


846 


862 


878 


894 


910 


927 


943 


959 


• 3 


4 


9 


4 


8 


269 


975 


991 


*007 


*023 


*040 


*056 *072 


=^088 


*104 


*120 


.4 
.5 


6 
8 


6 
2 


6 
8 


4 



270 


43 136 


152 


168 


184 


200 


216 


233 


249 


265 


281 


.6 
• 7 
■ 8 


9 
11 
13 


9 
5 
2 


9 
11 
12 


6 
2 
8 


271 


297 


313 


329 


345 


361 


377 


393 


409 


425 


441 


272 


457 


473 


489 


505 


520 


536 


552 


568 


584 


600 


.9 


14 


8 


14 


4 


273 


616 


632 


648 


664 


680 


695 


711 


727 


743 


759 




274 


775 


791 


806 


822 


838 


854 


870 


886 


901 


917 




275 


933 


949 


965 


980 


996 


^012 


-^028 


*043 


*059 


*075 




276 


44 091 


106 


122 


138 


154 


169 


185 


201 


216 


232 




277 


248 


263 


279 


295 


310 


326 


342 


357 


373 


389 




278 


404 


420 


435 


451 


467 


482 


498 


513 


529 


545 


15 15 


279 


560 


576 


591 


607 


622 


638 


653 


669 


685 


700 


.1 

.2 


1.5 
3.1 


1.5 
3.0 


280 


716 


731 


747 


762 


778 


793 


809 


824 


839 


855 


.3 


4.6 


4.5 
























. 


.4 


6-2 


6.0 


281 


870 


886 


901 


917 


932 


948 


963 


978 


994 


*009 


.5 


7.7 


7.5 


282 


45 025 


040 


055 


071 


086 


102 


117 


132 


148 


163 


• 6 


9.3 


9.0 


283 


178 


194 


209 


224 


240 


255 


270 


286 


301 


316 


•7 


10.8 


10.5 


284 


332 


347 


362 


377 


393 


408 


423 


438 


454 


469 


• 8 


12.4 


12.0 


285 


484 


499 


515 


530 


545 


560 


576 


591 


606 


621 


• 9 


13.9 


13.5 


286 


636 


652 


667 


682 


697 


712 


727 


743 


758 


773 




287 


788 


803 


818 


833 


848 


864 


879 


894 


909 


924 




288 


939 


954 


969 


984 


999 


^^014 


*029 


*044 


*059 


*075 




289 


46 090 


105 


120 


135 


150 


165 


180 


195 


210 


225 




290 


240 


255 


269 


284 


299 


314 


329 


344 


359 


374 


12 14 

























.1 


1.4 


1.4 


291 


389 


404 


419 


434 


449 


464 


479 


493 


508 


523 


• 2 


2.9 


2.8 


292 


538 


553 


568 


583 


597 


612 


627 


642 


657 


672 


.3 


4.3 


4.2 


293 


687 


701 


716 


731 


746 


761 


775 


790 


805 


820 


.4 


5.8 


5-6 


294 


834 


849 


864 


879 


894 


908 


923 


938 


952 


967 


.5 


7.2 


7.0 


295 


982 


997 


*011 


*026 


*041 


*055 


*070 


^085 


*100 


*114 


.6 


8.7 


8.4 


296 


47 129 


144 


158 


173 


188 


202 


217 


232 


246 


261 


.7 


10.1 


9.8 


297 


275 


290 


305 


319 


334 


348 


363 


378 


392 


407 


.8 


11.6 


11.2 


298 


421 


436 


451 


465 


480 


494 


509 


523 


538 


552 


.9 


13.0 


12.6 


299 


567 


581 


596 


610 


625 


639 


654 


668 


683 


697 




300 


712 


726 

1 


741 
2 


755 
3 


770 
4 


784 
5 


799 
6 


813 

7 


828 

8 


842 
9 




N. 







P, 


I 







572 









TABLE v.— LOGARITHMS OF NUMBERS. 










N. 





1 
726 


2 

741 


3 

755 


4 

770 


5 

784 


6 

799 


7 
813 


8 
828 


9 

842 


P. P. 


300 


47 712 




301 


856 


871 


885 


9O0 


914 


928 


943 


957 


972 


986 




302 


48 000 


015 


029 


044 


058 


072 


087 


101 


115 


130 


• 


303 


144 


158 


173 


187 


201 


216 


230 


244 


259 


273 




304 


287 


301 


316 


330 


344 


358 


373 


387 


401 


415 




305 


430 


444 


45£ 


472 


487 


501 


515 


529 


543 


558 




306 


572 


586 


600 


614 


62C 


643 


657 


671 


685 


699 


4 rr t M 


307 
308 


714 
855 


728 
869 


742 
883 


756 
897 


770 
9ll 


784 

925 


798 
939 


812 
953 


827 
967 


841 
982 


.1 
.2 
.3 


1.4 


x-* 

1.4 


309 


996 


*010 


*024 


*038 


*052 


*066 


*080 


*094 


*108 


*122 


4 


• 3 


4 


•0 

■ 2 


310 


49 136 
276 


150 
290 


164 
304 


178 
318 


192 
332 


206 
346 


220 
359 


234 
373 


248 


262 


.4 
.5 
.6 
• 7 
■ 8 


5 

7 

8 

10 

11 


8 

• 2 
■ 7 
1 

.6 


5 
7 
8 
9 

11 


•8 



311 


387 


401 


•4 
8 
2 


312 


415 


429 


443 


457 


471 


485 


499 


513 


526 


540 


313 


554 


568 


582 


596 


610 


624 


637 


651 


665 


679 


314 


693 


707 


729 


734 


748 


762 


776 


789 


803 


817 


.y.io uii^ -0 


315 


831 


845 


858 


872 


886 


900 


913 


927 


941 


955 




316 


968 


982 


996 


^010 


^023 


^037 


^051 


^065 


-078 


*092 




317 


50 106 


119 


133 


147 


160 


174 


188 


20:. 


215 


229 




318 


242 


256 


270 


283 


297 


311 


324 


338 


352 


365 




319 


379 


392 


406 


420 


433 


447 


460 


474 


488 


501 




330 


515 


528 


542 


555 


569 


583 


596 


610 


623 


637 




321 


650 


664 


677 


691 


704 


718 


731 


745 


758 


772 


• 1 
.2 
.3 

• 4 

• 5 

• 6 

• 7 
.8 

9 


15. 13 

-1 n 1 1 n 


322 


785 


799 


812 


826 


839 


853 


866 


880 


893 


907 


2 
4 
5 
6 
8 
9. 
10. 

^9,. 




7 

4 

7 
1 
4 

? 


2 
3 
5 
6 
7 
9 
10 
11 


<3 

6 
9 
2 
5 
8 
1 
4 
7 


323 
324 


920 
51 054 


933 
068 


947 
081 


960 
094 


974 
108 


987 

121 


=^^001 
135 


*014 
148 


*027 
161 


*041 
175 


325 


188 


201 


215 


228 


242 


255 


268 


282 


295 


308 


326 


322 


335 


348 


361 


375 


388 


401 


415 


428 


441 


327 
328 


455 
587 


468 
600 


481 
614 


494 
627 


508 
640 


521 
653 


534 
667 


547 
680 


561 
693 


574 
706 


329 


719 


733 


746 


759 


772 


785 


798 


812 


825 


838 


330 


851 


864 


877 


891 


904 


917 


930 


943 


956 


969 




331 


983 


996 


^009 


*022 


*035 


^048 


*06l 


*074 


*087 


=^100 




332 


52 11^: 


127 


140 


153 


166 


179 


192 


205 


218 


23 




333 


24^ 


257 


270 


283 


296 


309 


322 


335 


348 


361 




334 


374 


387 


400 


413 


426 


439 


452 


465 


478 


491" 




335 


50^: 


517 


530 


543 


556 


569 


582 


595 


608 


621 




336 


634 


647 


660 


672 


685 


698 


711 


724 


737 


750 


_ 


337 


763 


776 


789 


801 


814 


827 


840 


853 


866 


879 


• 1 

.2 
3 


! - 


LfC 


338 


891 


904 


917 


930 


943 


956 


968 


981 


994 


*007 


1 

2 
3 


'^ 


1 " 


■^ 


339 


53 020 


033 


045 


058 


071 


084 


097 


109 


122 


135 


^ 


2 
3 


4 
6 


340 


148 


160 


173 


186 


199 


211 


224 


237 


250 


262 


.4 
.5 


5 
6 




9 


4 
B 


8 



341 


275 


288 


301 


313 


326 


339 


352 


364 


377 


390 


• 6 
•7 

8 

• 9 


7 

8 

10 


5 
7 



7 
8 
9 


2 


342 


402 


415 


428 


440 


453 


466 


478 


491 


504 


516 


4 


343 


529 


542 


554 


567 


580 


592 


605 


618 


630 


643 


6 


344 


656 


668 


681 


693 


706 


719 


731 


744 


756 


769 


11 z 


io.» 


345 


782 


794 


807 


819 


832 


845 


857 


870 


882 


895 




346 


907 


920 


932 


945 


958 


970 


983 


995 


*008 


*020 




347 


54 033 


045 


058 


070 


083 


095 


108 


120 


133 


145 




348 


158 


170 


183 


195 


208 


220 


232 


245 


257 


270 




349 


282 


295 


307 


320 


332 


344 


357 


369 


382 


394 




350 


407 


419 


431 
2 


444 
3 


456 
4 


469 
5 


481 
6 


493 

7 


506 

8 


518 
9 




N. 





1 


P. P. 



573 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 
419 


2 

43i 


3 

444 


4 

456 


5 

469 


6 

481 


7 
493 


8 


9 


P 


. P. 


350 


54 407 


506 


518 




































Ig 


351 


530 


543 


555 


568 


580 


592 


605 


617 


629 


642 


.1 


1.2 


352 


654 


666 


679 


691 


703 


716 


728 


740 


753 


765 


.3 


2- 


5 


353 


777 


790 


802 


814 


826 


839 


851 


863 


876 


888 


.3 


3. 


7 


354 


900 


912 


925 


937 


949 


961 


974 


986 


998 *010 


• 4 


5. 





355 


55 023 


035 


047 


059 


071 


084 


096 


108 


120 


133 


.5 


6. 


2 


356 


145 


157 


169 


181 


194 


206 


218 


230 


242 


254 


.6 


7. 


5 


857 


267 


279 


291 


303 


315 


327 


340 


352 


364 


376 


.7 


8. 


7 


358 


388 


400 


412 


424 


437 


449 


461 


473 


485 


497 


■ 8 


10. 





359 


509 


521 


533 


545 


558 


570 


582 


594 


606 


618 


.9 


11. 


2 


360 


630 


642 


654 


666 


678 


690 


702 


714 


726 


738 




12 

12 


361 


750 


762 


775 


787 


799 


811 


823 


835 


847 


859 


.1 


362 


871 


883 


895 


907 


919 


931 


943 


955 


966 


978 


-2 


2 


4 


363 


990 


*002 


*014 


*026 


*038 


*050 


*062 


*074 


*086 


*098 


.3 


3 


6 


364 


56 110 


122 


134 


146 


158 


170 


181 


193 


205 


217 


.4 


4 


8 


365 


229 


241 


253 


265 


277 


286 


300 


312 


324 


336 


.5 


6 





366 


348 


360 


372 


383 


395 


407 


419 


431 


443 


455 


.6 


7 


2 


367 


466 


478 


490 


502 


514 


525 


537 


549 


561 


573 


.7 


8 


4 


368 


585 


596 


608 


620 


632 


643 


655 


667 


679 


691 


.8 


9 


6 


369 


702 


714 


726 


738 


749 


761 


773 


785 


796 


808 


.9 


10. 8 


370 


820 


832 


843 


855 


867 


879 


89D 


902 


914 


925 




J^i 


371 


937 


949 


961 


972 


984 


996 


*007 


*019 


*031 


*042 


.1 


372 


57 054 


066 


077 


089 


101 


112 


124 


136 


147 


159 


.2 


2 


3 


373 


171 


182 


194 


206 


217 


229 


240 


252 


264 


275 


.3 


3 


4 


374 


287 


299 


310 


322 


333 


345 


357 


366 


380 


391 


.4 


4 


6 


375 


403 


414 


426 


438 


449 


461 


472 


484 


495 


507 


.5 


5 


7 


376 


519 


530 


542 


553 


565 


576 


588 


599 


611 


622 


• 6 


6 


9 


377 


634 


645 


657 


668 


680 


691 


703 


714 


726 


737 


.7 


8 





378 


749 


760 


772 


783 


795 


806 


818 


829 


841 


852 


.8 


9 


2 


379 


864 


875 


887 


898 


909 


921 


932 


944 


955 


967 


.9 


10. 3 


380 


978 


990 


*001 


*012 


*024 


*035 


*047 


*058 


*069 


*081 




11 

11 


381 


58 092 


104 


115 


126 


138 


149 


161 


172 


183 


195 


.1 


382 


206 


217 


229 


240 


252 


263 


274 


286 


297 


308 


.2 


2 


2 


,^83 


320 


331 


342 


354 


365 


376 


388 


399 


410 


422 


3 


3 


3 


384 


433 


444 


455 


467 


478 


489 


501 


512 


523 


535 


• 4 


4 


■ 4 


385 


546 


557 


568 


580 


591 


602 


613 


625 


636 


647 


.5 


b 


5 


386 


658 


670 


681 


692 


703 


715 


726 


737 


748 


760 


.6 


6 


• 6 


387 


771 


782 


793 


804 


816 


827 


838 


849 


861 


872 


.7 


y 


7 


388 


883 


894 


905 


916 


928 


939 


950 


961 


972 


984 


.8 


8 


8 


389 


995 


*006 


*017 


*028 


*039 


*050 


*062 


=^073 


*084 


■^095 


.9 


99 


390 


59 106 


117 


128 


140 


151 


162 


173 


184 


195 


206 




n 


391 


217 


229 


240 


251 


262 


273 


284 


295 


306 


317 


.1 


392 


328 


339 


351 


362 


373 


384 


395 


406 


417 


428 


.2 


2 


i 


393 


439 


450 


46 


472 


483 


494 


505 


516 


527 


538 


.3 


3 


394 


549 


560 


57 


582 


593 


604 


615 


626 


637 


648 


.4 


4 


2 


395 


659 


670 


68 


692 


703 


714 


725 


736 


747 


758 


.5 


5 


2 


396 


769 


780 


791 


802 
912 


813 


824 


835 


846 


857 


868 


.6 


6 


3 


897 


879 


890 


901 


923 


933 


944 


955 


966 


977 


.7 


7 


5 


398 


988 


999 


*010 


*021 


*032 


*043 


*053 


*064 


*075 


*086 


.8 


8 


i 


399 


60 097 


108 


119 


130 


141 


151 


162 


173 


184 


195 


.9 


9.4 


400 


206 


217 

1 


227 
3 


238 
3 


249 
4 


260 
5 


271 
6 


282 
7 


293 
8 


303 
9 






N. 





P 


.P. 



574 









TABLE v.— LOGARITHMS OF NUMBERS. 






N. 





1 


2 


3 


4 


5 


6 


7 

282 
390 


8 

293 
401 


9 

303 
412 


P.P. 


400 

401 


60 206 


217 


227 


238 


249 


265 


271 






314 


325 


336 


347 


357 


368 


379 




402 


422 


433 


444 


455 


466 


476 


487 


498 


509 


519 


• 




403 


530 


541 


552 


563 


573 


584 


595 


606 


616 


627 


11 




404 


638 


649 


659 


670 


681 


692 


702 


713 


724 


735 


.1 


1.1 




405 


745 


756 


767 


777 


788 


799 


810 


820 


831 


842 


.2 


2.2 




406 


852 


863 


874 


884 


895 


906 


916 


927 


938 


949 


• 3 


3.3 




407 


959 


970 


981 


991 


*002 


*013 


*023 *034 


*044 


*055 


• 4 


4.4 




408 


61 066 


076 


087 


098 


108 


119 


130 


140 


151 


16l 


.5 


5.5 




409 


172 


183 


193 


204 


215 


225 


236 


246 


257 


268 


• 6 

• 7 
.8 


66 
7.7 
88 




410 


278 


289 


299 


310 


320 


331 


342 


352 


363 


373 


























.9 


9.9 


















411 


384 


394 


405 


416 


426 


437 


447 


458 


468 


479 






412 


489 


500 


511 


521 


532 


542 


553 


563 


574 


584 






413 


5a5 


605 


616 


626 


637 


647 


658 


668 


679 


689 






414 


700 


710 


721 


731 


742 


752 


763 


773 


784 


794 






415 


805 


815 


825 


836 


846 


857 


867 


878 


888 


899 


10 




416 


909 


920 


930 


940 


951 


961 


972 


982 


993 


*003 


.1 


1.0 




417 


62 013 


024 


034 


045 


055 


065 


076! 086 


097 


107 


.2 


2.1 




418 


117 


128 


138 


149 


159 


169 


180i 190 


200 


211 


.3 


3.1 




419 
420 


22l 


232 


242 


252 


263 


273 


283! 294 


304 


314 


.4 
• 5 
.6 


4-2 
5-2 
6.3 




325 


335 


345 


356 


366 


376 


387 397 


407 


418 
























.7 


7-3 
























421 


428 


438 


449 


459 


469 


480 


490 500 


510 


521 


.8 


8.4 




422 


531 


54l 


552 


562 


572 


582 


593 603 


613 


624 


.9 


9.i 




423 


634 


644 


654 


665 


675 


685 


695 706 


716 


726 






424 


736 


747 


757 


767 


777 


788 


798! 808 


818 


828 






425 


839 


849 


859 


86 


879 


890 


900 910 


920 


931 






426 


941 


951 


96l 


97 


981 


992 


*002*012 


*022 


*032 






427 


63 043 


053 


063 


073 


083 


093 


104 114 


124 


134 


10 




428 


144 


154 


164 


175 


185 


195 


2051 215 


225 


235 


.1 


1.0 




429 
430 

431 


245 


256 


266 


276 


286 


296 


306; 316 


326 


336 


.2 
.3 
.4 
.5 
.6 


2.0 
30 
4.0 
5.0 
6.0 




347 


357 


367 


377 


387 


397 


407 


417 


427 


437 




447 


458 


468 


478 


488 


498 


508 


518 


528 


538 




432 


548 


558 


568 


578 


588 


598 


6081 618 


628 


639 


.7 


7.0 




433 


649 


659 


669 


679 


689 


699 


709 719 


729 


739 


.8 


80 




434 


749 


759 


769 


779 


789 


799 


809' 819 


829 


839 


.9 


9.0 




435 


849 


859 


869 


879 


889 


899 


909 919 


928 


938 






436 


948 


958 


968 


978 


988 


998 


*008:*018 


*028 


*038 






437 


64 048 


058 


068 


078 


088 


098 


107 


117 


127 


137 






438 


147 


157 


167 


177 


187 


197 


207 


217 


226 


236 






439 


246 


256 


266 


276 


286 


296 


306 


315 


325 


335 


9_ 


























.1 


o.g 








440 


345 


355 


365 


375 


384 


394 


404 


414 


424 


434 


.2 
.3 


i.l 




























441 


444 


453 


463 


473 


483 


493 


503 


512 


522 


532 


.4 


3.8 




442 


542 


552 


562 


571 


581 


591 


601 


611 


621 


630 


.5 


4.7 




443 


640 


650 


660 


670 


679 


689 


699 


709 


718 


728 


.6 


5.7 




444 


738 


748 


758 


767 


777 


787 


797 


806 


816 


826 


.7 


6.6 




445 


836 


846 


855 


865 


875 


885 


894 


904 914 


923 


.8 


7.6 




446 


933 


943 


953 


962 


972 


982 


992*001 *0ll*021 


.9 


8.5 




447 


65 031 


040 


050 


060 


069 


079 


089 098! 108; 118 






448 


128 


137 


147 


157 


166 


176 


186 195; 205 215 






449 
450 


224 
321 


234 


244 


253 


263 


273 


282 292| 302 3ll 






331 


340 


350 


360 


369 


379 
6 


! 389 

7 


398 
8 


408 
9 




N. 





1 


2 


3 


4 


5 


P.P. 














575 













TABLE \ 


.—LOGARITHMS OF NUMBERS. 






N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


r 


. P. 


450 


65 32l 


331 


340 


350 


360 


369 


379 


389 


398 


408 






451 


417 


427 


437 


446 


456 


466 


475 


485 


494 


504 






452 


514 


523 


533 


542 


552 


562 


571 


581 


590 


600 




10 


453 


610 


619 


629 


638 


648 


657 


667 


677 


686 


696 


• 1 


10 


454 


705 


715 


724 


734 


744 


753 


763 


772 


782 


791 


• 2 


2 


• 


455 


801 


810 


820 


830 


839 


849 


858 


868 


877 


887 


• 3 


3 


• 


456 


896 


906 


915 


925 


934 


944 


953 


963 


972 


982 


• 4 


4 





457 


991 


^^001 


^010 


=*=020 


=^029 


=^039 


*048 


*058 


*067 


*077 


5 


rt 


.0 


458 


66 086 


096 


105 


115 


124 


134 


143 


153 


]62 


172 


• 6 


6 


■ 


459 


181 


190 


200 


209 


219 


228 


238 


247 


257 


266 


7 


7 


.0 
























8 


8 





460 


276 


28b 


294 


304 


313 


323 


332 


342 


351 


360 


9 


9 


.0 


461 


370 


379 


389 


398 


408 


417 


426 


436 


445 


455 




462 


464 


473 


483 


492 


502 


511 


520 


530 


539 


548 






463 


558 


567 


577 


586 


595 


605 


614 


623 


633 


642 






464 


652 


661 


670 


680 


689 


698 


708 


717 


726 


736 






465 


745 


754 


764 


773 


782 


792 


801 


810 


820 


829 




9 


466 


838 


848 


857 


86R 


876 


885 


894 


904 


913 


922 


.1 


0.9 


467 


931 


941 


950 


959 


969 


978 


987 


996 


*006 


*015 


■ 2 


1 


9 


468 


67 024 


034 


043 


052 


061 


071 


080 


089 


099 


108 


.3 


2 


• 8 


469 


117 


126 


136 


145 


154 


163 


173 


182 


191 


200 


• 4 


3 


8 



























• 5 


4 


•7 


470 


210 


219 


228 


237 


246 


256 


265 


274 


283 


293 


• 6 


5 


7 
































•7 


6 


6 


471 


302 


311 


320 


329 


339 


348 


357 


366 


376 


385 


8 


7 




472 


394 


403 


412 


422 


431 


440 


449 


458 


467 


477 


9 


8 


5 


473 


486 


495 


504 


513 


523 


532 


541 


550 


559 


568 






474 


578 


587 


596 


605 


614 


623 


633 


642 


651 


660 






475 


669 


678 


687 


697 


706 


715 


724 


733 


742 


751 






476 


760 


770 


779 


788 


797 


806 


815 


824 


833 


842 






477 


852 


861 


870 


879 


888 


897 


906 


915 


924 


933 






478 


943 


952 


96 . 


970 


979 


988 


997 


=^006 


*015 


*024 




9 


479 


68 033 


042 


051 


060 


070 


079 


088 


097 


106 


115 


• 1 

• 2 

• 3 




1 
2 


9 
8 

7 


480 


124 


133 


142 


151 


160 


169 


178 


187 


196 


205 





























•4 


3 


6 


481 


214 


223 


232 


241 


250 


259 


268 


277 


286 


295 


• 5 


4. 


5 


482 


304 


313 


322 


331 


340 


349 


358 


367 


376 


385 


• 6 


5. 


4 


483 


394 


403 


412 


421 


430 


439 


448 


457 


466 


475 


7 


6. 


3 


484 


484 


493 


502 


511 


520 


529 


538 


547 


556 


565 


.8 


7. 


2 


485 


574 


583 


592 


601 


610 


619 


628 


637 


646 


654 


9 


8. 


1 


486 


663 


672 


681 


690 


699 


708 


717 


726 


735 


744 






487 


753 


762 


770 


779 


788 


797 


806 


815 


824 


833 






488 


842 


851 


860 


868 


877 


886 


895 


904 


913 


922 






489 


931 


940 


948 


957 


966 


975 


984 


993 


=^002 


=^010 






490 


69 019 


028 


037 


046 


055 


064 


073 


081 


090 


099 




0-8 i 
























•1 


491 


108 


117 


126 


134 


143 


152 


161 


170 


179 


187 


• 2 


1. 


7 


492 


196 


205 


214 


223 


232 


240 


249 


258 


267 


276 


• 3 


2- 


5 


493 


284 


293 


302 


311 


320 


328 


337 


346 


355 


364 


.4 


3- 


4 


494 


372 


381 


390 


399 


408 


416 


42^ 


434 


443 


451 


• 5 


4. 


2 


495 


460 


469 


478 


487 


495 


504 


513 


522 


530 


539 


.6 


5. 


1 


496 


548 


557 


565 


574 


583 


592 


600 


609 


618 


627 


.7 


5. 


9 ^ 


497 


635 


644 


653 


662 


670 


679 


688 


697 


705 


714 


• 8 


6- 




498 


723 


731 


740 


749 


758 


766 


775 


784 


792 


801 


• 9 


7. 


6 ' 


499 


810 
897 


819 
905 

1 


827 
914 

2 


836 


845 


853 


862 


871 


879 
966 

8 


888 

975 

9 




1 

J 


600 


923 
3 


93l 
4 


940 
5 


949 
6 


958 

7 


N. 





P. 


P. 
















576 



















TABLE V 


.—LOGARITHMS OF NUMBERS. 






N. 





1 
905 


2 

914 


3 

923 


4 

931 


5 


6 

949 


7 
958 


8 
966 


9 

975 


P. P. 




500 


69 897 


940 






501 


984 


992 


*001 


*010 


*018 


*027 


*036 


*044 


*053 


*06l 






502 


70 070 


079 


087 


096 


105 


113 


122 


131 


139 


148 


9 




503 


157 


165 


174 


182 


191 


200 


208 


217 


226 


234 


.1 


0.9 




504 


243 


251 


260 


269 


277 


286 


294 


303 


312 


320 


.2 


1.8 




505 


329 


337 


346 


355 


363 


372 


380 


389 


398 


406 


• 3 


2.7 




506 


415 


423 


432 


441 


449 


458 


466 


475 


483 


492 


-4 


36 




507 


501 


509 


518 


526 


535 


543 


552 


560 


569 


578 


• 5 


4-5 




508 


586 


595 


603 


612 


620 


629 


637 


646 


654 


663 


• 6 


5.4 




509 


672 


680 


689 


697 


706 


714 


723 


731 


740 


748 


• 7 
8 

• 9 


6-3 
7-2 
8.1 




510 


757 


765 


774 


782 


791 


799 


808 


816 


825 


833 




511 


842 


850 


859 


867 


876 


884 


893 


901 


910 


918 






512 


927 


935 


944 


952 


961 


969 


978 


986 


995 


*003 






513 


71 Oil 


020 


028 


037 


045 


054 


062 


071 


079 


088 






514 


096 


105 


113 


121 


130 


138 


147 


155 


164 


172 






515 


180 


189 


197 


206 


214 


223 


231 


239 


248 


256 


^ 8 




516 


265 


273 


282 


290 


298 


307 


315 


324 


332 


340 


• If 


08 




517 


349 


357 


366 


374 


382 


391 


399 


408 


416 


424 


■2 


1.7 




518 


433 


441 


449 


458 


466 


475 


483 


491 


500 


508 


.3 


2-5 




519 


516 


525 


533 


542 


550 


558 


567 


575 


583 


592 


•4 
.5 


34 

4.2 




























520 


600 


608 


617 


625 


633 


642 


650 


659 


667 


675 


• 6 


5-1 
































• 7 


5.9 




521 


684 


692 


700 


709 


717 


725 


734 


742 


750 


758 


.8 


68 




522 


767 


775 


783 


792 


800 


808 


817 


825 


833 


842 


.9 


7.6 




523 


850 


858 


867 


875 


883 


891 


900 


908 


916 


925 






524 


933 


941 


949 


958 


966 


974 


983 


991 


999 


*007 






525 


72 016 


024 


032 


040 


049 


057 


065 


074 


082 


090 






526 


098 


107 


115 


123 


131 


140 


148 


156 


164 


173 






527 


181 


189 


197 


206 


214 


222 


230 


238 


247 


255 






528 


263 


271 


280 


288 


296 


304 


312 


321 


329 


337 


8 




529 


345 


354 


362 


370 


378 


386 


395 


403 


411 


419 


.1 


0.8 
1.6 

2.4 




530 


427 


436 


444 


452 


460 


468 


476 


485 


493 


501 


• 3 
































.4 


3-2 




531 


509 


517 


526 


534 


542 


5bO 


558 


566 


575 


583 


• 5 


4.0 




532 


591 


599 


607 


615 


624 


632 


640 


648 


656 


664 


.6 


48 




533 


672 


681 


689 


697 


705 


713 


721 


729 


738 


746 


•7 


5-6 




534 


754 


762 


770 


778 


786 


795 


803 


811 


819 


827 


• 8 


6-4 




535 


835 


843 


851 


859 


868 


876 


884 


892 


900 


908 


.9 


7.2 




536 


916 


924 


932 


941 


949 


957 


965 


973 


981 


989 






537 


997 


=^005 


*013 


*021 


*030 


*038 


*046 


*054 


*062 


*070 






538 


73 078 


086 


094 


102 


110 


118 


126 


134 


143 


151 






539 


159 


167 


175 


183 


191 


199 


207 


215 


223 


231 






540 


239 


247 


255 


263 


27l 


279 


287 


295 


303 


311 


.1 

.2 


0^ 

1.5 




541 


319 


328 


336 


344 


352 


360 


368 


376 


384 


392 




542 


400 


408 


416 


424 


432 


440 


448 


456 


464 


472 


• 3 


2.2 




543 


480 


488 


496 


504 


512 


520 


528 


536 


544 


552 


.4 


30 




544 


560 


568 


576 


584 


592 


600 


608 


615 


623 


631 


.5 


3-7 




545 


639 


647 


655 


663 


671 


679 


687 


695 


703 


711 


.6 


4.5 




546 


719 


727 


735 


743 


751 


759 


767 


775 


783 


791 


.7 


5-2 




547 


798 


806 


814 


822 


830 


838 


846 


854 


862 


870 


.8 


6 




548 


878 


886 


894 


902 


909 


917 


925 


933 


941 


949 


9 


6.7 




549 


957 


965 


973 


981 


989 


997 


*004 


*012 


*020 


*028 






550 


74 036 


044 


052 


060 


068 


075 


083 


091 

7 


099 

8 


107 
9 






N. 





1 


2 


3 


4 


5 


6 


P. P. 



577 









TABLE v.— LOGARITHMS OF NUMBERS. 




i 


N. 





1 


2 


3 


4 

068 


5 

075 


6 

083 


7 
091 


8 

099 


9 

107 


P. P. 


550 


74 036 


044 


052 


060 




551 


115 


123 


131 


139 


146 


154 


162 


170 


178 


186 




552 


194 


202 


209 


217 


225 


233 


241 


249 


257 


264 




553 


272 


280 


288 


296 


304 


312 


319 


327 


335 


343 




554 


351 


359 


366 


374 


382 


390 


398 


406 


413 


421 




555 


429 


437 


445 


453 


460 


468 


476 


4:84 


492 


499 




556 


507 


515 


523 


531 


538 


546 


554 


562 


570 


577 




557 


585 


593 


601 


609 


616 


624 


632 


640 


648 


655 


.1 

• 2 

• 3 
.4 
.5 


» 


558 


663 


671 


679 


687 


694 


702 


710 


718 


725 


733 





I 


559 


741 
819 


749 
826 


756 
834 


764 
842 


772 
850 


780 
857 


788 
865 


795 


803 


811 


1 
2 
3 

1 4 


6 

4 
2 



560 


873 


881 


888 


561 
562 


896 
973 


904 
981 


912 
989 


919 
997 


927 
*004 


935 
*012 


942 
*020 


950 
*027 


958 
*035 


966 
*043 


• 6 
.7 
•8 
.9 


4 
5 
6 


8 

6 
4 


563 


75 051 


058 


066 


074 


08 


089 


097 


105 


112 


120 


564 


128 


135 


143 


151 


158 


166 


174 


182 


189 


197 


7 -A 


565 


205 


212 


220 


228 


235 


243 


251 


258 


266 


274 




566 


281 


289 


297 


304 


312 


320 


327 


335 


343 


350 




567 


358 


366 


373 


381 


389 


396 


404 


412 


419 


427 




568 


435 


442 


450 


458 


465 


473 


480 


488 


496 


503 




569 


511 


519 


526 


534 


541 


549 


557 


564 


572 


580 




570 


587 


595 


602 


610 


618 


625 


633 


641 


648 


656 


1 


571 
572 
573 


663 
739 
815 


671 
747 
823 


679 
755 
830 


686 
762 
838 


694 
770 
846 


70l 
777 
853 


709 
785 
861 


717 
792 
868 


724 
800 
876 


732 
808 
883 


.1 
.2 
.3 

• 4 

• 5 

1 

8 
.9 


o' 

1 

2 
3 
3 
4 
5 
6 
6 


r 

7 


7 

i 


574 


891 


899 


906 


914 


921 


929 


936 


944 


951 


959 


575 


967 


974 


982 


989 


997 


=^004 


*012 


*019 


*027 


=^'034 


576 
577 


76 042 
117 


050 
125 


057 
132 


065 
140 


072 
147 


080 
155 


087 
162 


095 
170 


102 
178 


110 
185 


678 


193 


200 


208 


215 


223 


230 


238 


245 


253 


260 


579 


268 


275 


283 


290 


298 


305 


313 


320 


328 


335 


580 


343 


350 


358 


365 


372 


380 


387 


395 


402 


410 




581 


417 


425 


432 


440 


447 


455 


462 


470 


477 


485 




582 


492 


500 


507 


514 


522 


529 


537 


544 


552 


559 




583 


567 


574 


582 


589 


596 


604 


611 


619 


626 


634 




584 


641 


648 


656 


663 


671 


678 


686 


693 


700 


708 




585 


715 


723 


730 


738 


745 


752 


760 


767 


775 


782 




586 


790 


797 


804 


812 


819 


827 


834 


841 


849 


856 


^ 


587 
588 
589 


864 

937 

77 Oil 


871 
945 
019 


878 
952 
026 


886 
960 
033 


893 
967 
041 


901 
974 
048 


908 
982 
055 


915 
989 
063 


923 
997 
070 


930 

*004 

078 


.2 
.3 


0.7 

1.4 
2.1 


590 


085 


092 


100 


107 


114 


122 


129 


136 


144 


151 


.4 
• 5 


28 
35 


591 


158 


166 


173 


181 


188 


195 


203 


210 


217 


225 


• 6 

• 7 
8 

.9 


4.2 
4.9 
5-6 
6.3 


592 
593 
594 


232 
305 
378 


239 
313 
386 


247 
320 
393 


254 
327 
400 


261 
335 
408 


269 
342 
415 


276 

34:9 

422 


283 
356 
430 


291 
364 
437 


298 
371 

444 


595 


451 


459 


466 


473 


481 


488 


495 


503 


510 


517 




596 


524 


532 


539 


546 


554 


561 


568 


575 


583 


590 




597 


597 


604 


612 


619 


626 


634 


641 


648 


655 


663 




598 


670 


677 


684 


692 


699 


706 


713 


721 


728 


735 




599 


742 
815 


750 
822 


757 
829 


764 


771 


779 


786 
858 

6 


793 
866 

4 


800 
873 

8 


808 
880 

9 




600 


837 


844 
4 


85l 
5 


N. 





1 


2 


3 


P. P. 














57 


8 












1 



tablp: v.— logarithms of numbers. 



N. 





1 


2 


3 

837 


4 

844 


5 

851 


6 

858 


7 
866 


8 
873 


9 

880 
952 


P 


. P. 


600 


77 815 
887 


822 


829 






601 


894 


902 


909 


916 


923 


931 


938 


945 




602 


959 


967 


974 


981 


988 


995 


*003 


*010 


*017 


*024 






603 


78 031 


039 


046 


053 


060 


067 


075 


082 


089 


096 






604 


103 


111 


118 


125 


132 


139 


147 


154 


161 


168 






605 


175 


182 


190 


197 


204 


211 


218 


226 


233 


240 






606 


247 


254 


261 


269 


276 


283 


290 


297 


304 


311 




2^2 


607 


319 


826 


333 


340 


347 


354 


362 


369 


376 


383 


.1 

.2 
.3 


608 


390 


397 


404 


412 


419 


426 


433 


440 


447 


454 


609 


461 


469 


476 


483 


490 


497 


504 


511 


518 


526 


610 


533 


540 


547 


554 


561 


568 


575 


583 


590 


597 


.4 
.5 


13 


611 


604 


611 


618 


625 


632 


639 


646 


654 


661 


668 


.6 
.7 
.8 
.9 


4.5 
5.2 

V7 


612 


675 


682 


689 


696 


703 


710 


717 


725 


732 


739 


613 


746 


753 


760 


767 


774 


781 


788 


795 


802 


810 


614 


817 


824 


831 


838 


845 


852 


859 


866 


873 


880 


615 


887 


894 


901 


908 


915 


923 


930 


937 


944 


951 






616 


958 


965 


972 


979 


986 


993 


*000 


*007 


*014 


*021 






617 


79 028 


035 


042 


049 


056 


063 


070 


078 


085 


092 






618 


099 


106 


113 


120 


127 


134 


141 


148 


155 


162 






619 


169 
239 


176 
246 


183 
253 


190 
260 


197 


204 


211 


218 


225 


232 






630 


267 


274 


281 


288 


295 


302 




621 


309 


316 


323 


330 


337 


344 


351 


358 


365 


372 


.1 

.2 
.3 

.4 
.5 
.6 
.7 
.8 
.9 


0^7 

1.4 
2.1 
2.8 
35 
4.2 
4.9 
5.6 
6.3 


622 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 


623 
624 


449 
518 


456 
525 


462 
532 


469 
539 


476 
546 


483 
553 


490 
560 


497 
567 


504 
574 


511 
581 


625 


588 


595 


602 


609 


616 


622 


629 


636 


643 


650 


626 
627 


657 
727 


664 
733 


740 


678 

747 


685 

754 


692 
761 


699 
768 


706 
775 


713 
782 


720 
789 


628 


796 


803 


810 


816 


823 


830 


837 


844 


851 


858 


629 


865 


872 


879 


886 


892 


899 


906 


913 


920 


927 


630 


934 


941 


948 


954 


96l 


968 


975 


982 


989 


996 






631 


80 003 


010 


016 


023 


030 


037 


044 


051 


058 


065 






632 


071 


078 


085 


092 


099 


106 


113 


120 


126 


133 






633 


140 


147 


154 


161 


168 


174 


181 


188 


195 


202 






634 


209 


216 


222 


229 


236 


243 


250 


257 


263 


270 






635 


277 


284 


291 


298 


304 


311 


318 


325 


332 


339 






636 


345 


352 


359 


366 


373 


380 


386 


393 


400 


407 




0^5 

1.3 
1-9 


637 


414 


421 


427 


434 


441 


448 


455 


461 


468 


475 


.1 
.2 
3 


638 
639 


482 
550 


489 

557 


495 
563 


502 
570 


509 
577 


516 
584 


523 
591 


529 
597 


536 
604 


543 
611 


640 


618 


625 


63l 


638 


645 


652 


658 


665 


672 


679 


.4 
.5 


2.6 
3.2 


641 


686 


692 


699 


706 


713 


719 


726 


733 


740 


746 


.6 
.7 
.8 
• 9 


39 
4.5 
5.2 
5.§ 


642 


753 


760 


767 


774 


780 


787 


794 


801 


807 


814 


643 
644 


821 
888 


828 
895 


834 
902 


841 
909 


848 
915 


855 

922 


861 
929 


868 
936 


875 
942 


882 
949 


645 


956 


962 


969 


976 


983 


989 


996 


*003 


*010 


*016 






646 


81 023 


030 


036 


043 


050 


057 


063 


070 


077 


083 






647 


090 


097 


104 


110 


117 


124 


130 


137 


144 


151 






648 


157 


164 


171 


177 


184 


191 


197 


204 


211 


218 






649 


224 


231 


238 


244 


251 


258 


264 


271 


278 


284 






650 


29l 


298 

1 


304 


311 


318 


324 


331 
6 


338 

7 


345 
8 


351 
9 






N. 





2 


3 


4 


5 


P 


. P. 



579 









lABLE V 


.—LOGARITHMS OF NUMBERb. 




1 


N. 





1 


2 


3 


4 


5 


6 

331 


7 

338 

405 


8 


9 

35l 


P 


P. n 


650 


81 29l 
358 


298 


304 


311 
378 


318 
385 


324 


345 

411 






651 


365 


371 


39l 


398 


418 


652 


425 


431 


438 


444 


451 


458 


464 


471 


478 


484 






653 


491 


498 


504 


511 


518 


524 


531 


538 


544 


551 






654 


558 


564 


571 


577 


584 


591 


597 


604 


611 


617 






655 


624 


631 


637 


644 


650 


657 


664 


670 


677 


684 






656 


690 


697 


703 


710 


717 


723 


730 


736 


743 


750 




m 


657 
658 
659 


756 
822 
888 


763 
829 
895 


770 
836 
901 


776 
842 
908 


783 
849 
915 


789 
855 
921 


796 
862 
928 


803 
869 
934 


809 
875 
941 


816 
882 
948 


• 1 

• 2 
.3 




1 

9 


7 
4 

1 


660 


954 

82 020 
086 
151 
217 


961 

026 
092 
158 

223 


967 

033 
099 
164 
230 


974 

040 
105 
171 
236 


980 

046 
112 
177 
243 


987 

053 
118 
184 
249 


994 

059 
125 
190 
256 


*0C0 

066 
131 
197 
262 


*007 

072 
138 

203 
269 


*013 


• 4 

• 5 

• 6 
•7 

8 

• 9 


2 
3 
4 
4 
5 
6 


8 

5 

2 

9 ' 

6 

3 


661 
662 
663 
664 


079 
145 
210 
275 


665 


282 


288 


295 


302 


308 


315 


321 


328 


334 


341 




' 


666 


347 


354 


360 


367 


373 


380 


386 


393 


399 


406 






667 


412 


419 


425 


432 


438 


445 


451 


458 


464 


471 




j 


668 


477 


484 


490 


497 


503 


510 


516 


523 


529 


536 






669 


542 


549 


555 


562 


568 


575 


581 


588 


594 


601 






670 


607 


614 


620 


627 


633 


640 


646 


653 


659 


666 






671 


672 


678 


685 


691 


698 


704 


711 


717 


724 


730 


•1 

• 2 
3 

• 4 

• 5 

• 6 

• 7 
8 

• 9 


6 


672 


737 


743 


750 


756 


763 


769 


775 


782 


788 


795 


1 
1 
2 
3 
3 
4 
5 
5 


u 

3 

9 


673 


801 


808 


814 


821 


827 


834 


840 


846 


853 


859 


674 


866 


872 


879 


885 


892 


898 


904 


911 


917 


924 


675 


930 


937 


943 


94S 


956 


962 


969 


975 


982 


988 


676 


994 


*001 


••=007 


^i^OU 


*020 


=^027 


*033 


*039 


*046 


*052 


677 
678 
679 


83 059 
123 
187 


065 
129 
193 


071 
136 
200 


078 
142 
206 


084 
148 
212 


091 
155 
219 


097 
161 
225 


103 
168 

231 


110 
174 
238 


116 
180 

244 


680 


251 


257 


263 


270 


276 


283 


289 


295 


302 


308 






681 


314 


321 


327 


334 


340 


346 


353 


359 


365 


372 






682 


378 


385 


391 


397 


404 


410 


416 


423 


429 


435 






683 


442 


448 


455 


461 


467 


474 


480 


486 


493 


499 






684 


505 


512 


518 


524 


531 


537 


543 


550 


556 


562 






685 


569 


575 


581 


588 


594 


600 


607 


613 


619 


626 






686 


632 


638 


645 


651 


657 


664 


670 


676 


683 


689 




6 


687 


695 


702 


708 


714 


721 


727 


733 


740 


746 


752 


1 

.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


688 


759 


765 


771 


778 


784 


790 


796 


803 


809 


815 


u 

1 
1 
2 
3 
3 
4 
4 


o 

2 
8 
4 


6 

o 


689 


822 
885 
948 


828 
891 
954 


834 
897 
960 


841 
904 
966 


847 
910 
973 


853 
916 
979 


859 


866 
929 


872 
935 
998 


878 


690 


922 


941 


691 


985 


992 


=^004 


692 
693 


84 010 
073 


017 
079 


023 
086 


029 
092 


035 
098 


042 
104 


048 

111 


054 
117 


061 
123 


067 

129 


8 


694 


136 


142 


148 


154 


161 


167 


173 


179 


186 


192 


•* 


695 


198 


204 


211 


217 


223 


229 


236 


242 


248 


254 






696 


261 


267 


273 


279 


286 


292 


298 


304 


311 


317 






697 


323 


329 


335 


342 


348 


354 


360 


367 


373 


379 






698 


385 


392 


398 


404 


410 


416 


423 


429 


435 


441 






699 


447 
510 


454 
516 


460 


466 


472 


479 


485 


491 


497 


503 




? 


700 


522 


528 


534 
4 


541 
5 


547 
6 


553 

7 


559 
8 


565 


N. 





1 


2 


3 


9 


P. 


P. 





580 









TABLE v.— LOGARITHMS OF NUMBERS. 






N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


r 


'. P. 


700 


84 510 
572 


516 
578 


522 


528 
590 


534 
596 


541 
603 


547 
609 


553 
615 


559 
62l 


565 
627 






701 


584 




702 


633 


640 


646 


652 


658 


664 


671 


677 


683 


689 






703 


695 


701 


708 


714 


720 


726 


732 


739 


745 


751 






704 


757 


763 


769 


776 


782 


788 


794 


800 


806 


813 






705 


819 


825 


831 


837 


843 


849 


856 


862 


868 


874 






706 


880 


886 


893 


899 


905 


911 


917 


923 


929 


936 




0^- 


707 


942 


948 


954 


960 


966 


972 


979 


985 


991 


997 


.1 

.2 
.3 


708 
709 


85 003 
064 


009 
070 


015 
077 


021 
083 


028 
089 


034 
095 


040 
lOl 


046 
107 


052 
113 


058 
119 


1 
1 


D 

3 
9 


710 


126 


132 


138 


144 


150 


156 


162 


168 


174 


181 


.4 
.5 


2 
8 




711 


187 


193 


199 


205 


211 


217 


223 


229 


236 


242 


• 6 
.7 

• 8 
.9 


3 

4 
5 


9 
5 

2 


712 


248 


254 


260 


266 


272 


278 


284 


290 


297 


303 


713 


309 


315 


321 


327 


333 


339 


345 


351 


357 


363 


714 


370 


376 


382 


388 


394 


400 


406 


412 


418 


424 




715 


430 


436 


443 


449 


455 


461 


467 


473 


479 


485 






716 


491 


497 


503 


509 


515 


521 


527 


533 


540 


546 






717 


552 


558 


564 


570 


576 


582 


588 


594 


600 


606 






718 


612 


618 


624 


630 


636 


642 


648 


655 


661 


667 






719 


673 


679 


685 


691 


697 


703 


709 


715 


721 


727 






730 


733 


739 


745 


751 


757 


763 


769 


775 


781 


787 




721 


793 


799 


805 


811 


817 


823 


829 


835 


841 


847 


.1 

• 2 
.3 
.4 
.5 
.6 

• 7 

• 8 
.9 


6 


722 


853 


859 


865 


872 


878 


884 


890 


896 


902 


908 


'^ 

1 

1 

2 

3 

3 

4 

4 

5 




2 
8 
4 

6 
2 
8 
4 


723 


914 


920 


926 


932 


938 


944 


950 


956 


962 


968 


724 
725 


974 
86 034 


980 
040 


986 
046 


992 
052 


998 
058 


*004 
068 


*010 
069 


*016 
075 


*022 
081 


*028 
087 


726 


093 


099 


105 


111 


117 


123 


129 


135 


141 


147 


727 


153 


159 


165 


171 


177 


183 


189 


195 


201 


207 


728 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 


729 


273 


278 


284 


290 


296 


302 


308 


314 


320 


326 


730 


332 


338 


344 


350 


356 


362 


368 


374 


380 


386 






731 


391 


397 


403 


409 


415 


421 


427 


433 


439 


445 






732 


451 


457 


463 


469 


475 


481 


486 


492 


498 


504 






733 


510 


516 


522 


528 


534 


540 


546 


552 


558 


563 






734 


569 


575 


581 


587 


593 


599 


605 


611 


617 


623 






735 


628 


634 


640 


646 


652 


658 


664 


670 


676 


682 






736 


688 


693 


699 


705 


711 


717 


723 


729 


735 


741 




_ 


737 


746 


752 


758 


764 


770 


776 


782 


788 


794 


800 


.1 

• 2 

• 3 


5. 


738 


805 


811 


817 


823 


829 


835 


841 


847 


852 


858 








739 


864 


870 


876 


882 


888 


894 


899 


905 


911 


917 


1 
1 


1 
6 


740 


923 


929 


935 


941 


946 


952 


958 


964 


970 


976 


•4 
• 5 


2 

9 


2 
7 


741 


982 


987 


993 


999 


*005 


*011 


*017 


*023 


*028 


*034 


.6 

• 7 

• 8 
.9 


3 
3 




742 


87 040 


046 


052 


058 


064 


069 


075 


081 


087 


093 


743 
744 


099 
157 


104 
163 


110 
169 


116 
175 


122 
180 


128 

186 


134 
192 


140 
198 


145 
204 


151 
210 


4 
4 


i 


745 


215 


221 


227 


233 


239 


245 


250 


256 


262 


268 






746 


274 


279 


285 


291 


297 


303 


309 


314 


320 


326 






747 


332 


338 


343 


349 


355 


361 


367 


372 


378 


384 






748 


390 


396 


402 


407 


413 


419 


425 


431 


436 


442 






749 


448 


454 


460 


465 


471 


477 


483 


489 


494 


500 






750 


506 


512 


517 
2 


523 
3 


529 
4 


535 
5 


541 
6 


546 

7 


552 


558 
9 






N. 





1 


8 


P 


. P. 



581 









TABLE v.— LOGARITHMS 


OF NUMBERS. 






N. 





1 2 


3 


4 


5 


6 


7 


8 


9 


P 


. P. 


750 


87 506 


512 


517 


523 


529 


535 


541 


546 


552 


558 






751 


564 


570! 575 


58l! 587 


593 


598 


604 


610 


616 






752 


622 


627 633 


639 645 


650 


656 


662 


668 


673 






753 


679 


685 691 


697 


702 


708 714 


720 


725 


731 






754 


737 


743 748 


754 


760 


766 77l 


777 


783 


789 






755 


794 


800 806 812 


817 


823 829 


835 


840 


846 






756 


852 


858 8631 869 


875 


881' 886| 892 


898 


904 




6 


757 


909 


915 921 927 


932 


938 944I 949 


955 


961 


.1 
.2 
.3 


758 


967 


972 978 984 


990 


995 *00i;*007 


*012 


*018 


" 

1 

1 


D 


759 


88 024 


030, 035; 041 


047 


053 


058 


064 


070 


075 


2 
8 


760 


081 


087 


093 


098 


104 


110 


115 


I2T 


127 


133 


.4 
.5 


2 
3 


4 



761 


138 


144 


150 


155 


161 


167 


172 


178 


184 


190 


.6 
• 7 
.8 
.9 


3 


6 
2 
8 


762 


195 


1 201 207 


212 


218 


224 


229 235 


241 


247 


4 
4 


763 


252 


258 264' 269 


275 


281 


286 292 


298 


303 


764 


309 


315 320 326 


332 


337 


343; 349 


355 


360 


5"% 


765 


366 


372 377 383 


389 


394 


400 406 


411 


417 






766 


423 


4281 434 440 


445 


451 


457i 462 


468 


474 






767 


479 


485 491 496 


502 


508 


513 519 


525 


530 






768 


536 


542, 5471 553 


558 


564 


570 575 


581 


587 






769 


595 


598| 604| 609 


615 


621 


626| 632 


638 


643 






770 


649 


654 660 666 


671 


677 


683 


688 


694 


700 






771 


705 


711 716 722 


728 


733 


739 


745 


750 


756 


.1 
.2 
.3 
.4 
.5 
• 6 
.7 
.8 
.9 


^' 


772 


761 


767i 773 778 


784 


790 


795 


801 


806 


812 




1 
1 
2 
2 
3 
3 




I 

2 
7 
3 
8 


773 


818 


823' 829 835 


840 


846 


85l 


857 


863 


868 


774 


874 


879 885 891 


896 


902 


907 


913 


919 


924 


775 


930 


936 94l 947 


952 


958 


964 


969 


975 


980 


776 


986 


992 997 *003 


*008 


^014*019 


*025 


=^031 


*036 


777 
778 


89 042 
098 


047 053, 059 
103 109' 114 


064 
120 


070 
126 


075 
131 


081 
137 


087 
142 


092 
148 


779 


153 


159| 165 


170 


176 


181 


187 


193 


198 


204 


4* 
4.9 


780 


209 


215 


220 


226| 


231 


237 


243 


248 


254 


259 






781 


265 


270 


276 


282' 


287 


293 


298 


304 


309 


315 






782 


320 


326 


332 


337 


343 


348 


354 


359 


365 


370 






783 


376 


381 


387 


393 


398 


404 


409 


415 


420 


426 






784 


431 


437 


442 


448 


454 


459 


465 


470 


476 


481 






785 


487 


492 


498 


503 


509 


514 


520 


525 


531 


536 






786 


542 


548 


553 


559 


564 


570 


575 


581 


586 


592 




5 


787 


597 


603 


608 


614 


619 


625 


630 


636 


641 


647 


.1 
.2 
.3 


788 
789 


652 
707 


658 
713 


663 
718 


669 
724 


674 
729 


680 
735 


685 
740 


691 
746 


696 
751 


702 
757 


U 

1 
1 



5 


790 


762 


768 


773 


779' 


784 


790 


795 


801 


806 


812 


.4 
.5 


2 
2. 



5 


791 


817 


823 


828 


834 


839 


845 


850 


856 


861 


867 


.6 
.7 
.8 
.9 


3. 
3. 

A 



5 

n 


792 


872 


878 


883 


889 


894 


900 


905 


911 


916, 


922 


793 


927 


933 


938 


943 


949 


954 


960 


965 


97ll 


976 


4 -u 
4.5 


794 


982 


987 


993 


998 


*004 


*009 *015 


*020 


*026 


*031 


795 


90 036 


042 


047 


053 


058 


064 


069 


075 


080| 


086 






796 


091 


097 


102 


107 


113 


118 


124 


129 


135^ 


140 






797 


146 


151 


156 


162 


167 


173 


178 


184 


189! 


195 






798 


205 


205 


211 


216 


222 


227 


233 


238 


244 


249 






799 


254 


260 


265 


271 


276 


282 


287 


292 


298' 


303 






800 


309 


314 


320i 


325 

! 


330 


336 
5 


341 
6 


347 

7 


352j 
8 


358 
9 






N. 





1 


2 


3 4 


1*. 


P. 
















582 















TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 

341 


7 
347 


8 

352 
406 


9 

358 

412 


P. P. 


800 


90 309 


314 


320 


325 


330 


336 




801 


363 


368 


374 


379 


385 


390 


396 


401 




802 


417 


423 


428 


433 


439 


444 


450 


455 


460 


466 




803 


471 


477 


482 


488 


493 


498 


504 


509 


515 


520 




804 


525 


531 


536 


542 


547 


552 


558 


563 


569 


574 




805 


579 


585 


590 


596 


601 


606 


612 


617 


622 


628 




806 


633 


639 


644 


649 


655 


660 


666 


671 


676 


682 




807 


687 


692 


698 


703 


709 


714 


719 


725 


730 


736 




808 


741 


746 


752 


757 


762 


768 


773 


778 


784 


789 




809 


795 
848 


800 


805 


811 


816 


821 


827 


832 


838 


843 




810 


854 


859 


864 


870 


875 


880 


886 


89l 


896 




811 
812 
813 
814 
815 
816 


902 
955 
91 009 
062 
116 
169 


907 
961 
014 
068 
121 
174 


913 
966 
019 
073 
126 
179 


918 
971 
025 
078 
131 
185 


923 
977 
030 
084 
137 
190 


929 
982 
036 
089 
142 
195 


934 
987 
041 
094 
147 
201 


939 
993 
046 
100 
153 
206 


945 
998 
052 
105 
158 
211 


950 
=^003 
057 
110 
163 
217 




1 
2 
3 
4 
5 
6 
7 
8 
9 


0^5 

1.1 

1.6 

i:f 

3.3 
3-8 


817 
818 
819 


222 
275 
328 


227 
280 
333 


233 
286 
339 


238 
291 

344 


243 
296 
349 


249 
302 
355 


254 
307 
360 


259 
312 
365 


264 
318 
371 


270 
323 
376 




820 


381 


386 


392 


397 


402 


408 


413 


418 


423 


429 




821 


434 


439 


445 


450 


455 


461 


466 


47l 


476 


482 




822 


487 


492 


497 


503 


508 


513 


519 


524 


529 


534 




823 


540 


545 


550 


556 


561 


566 


571 


577 


582 


587 




824 


592 


598 


603 


608 


614 


619 


624 


629 


635 


640 




825 


645 


650 


656 


661 


666 


671 


677 


682 


687 


69^ 




826 


698 


703 


708 


714 


719 


724 


729 


735 


740 


745 




827 


750 


756 


761 


766 


771 


777 


782 


787 


792 


798 




828 


803 


808 


813 


819 


824 


829 


834 


839 


845 


850 




829 


855 


860 


866 


871 


876 


881 


887 


892 


897 


902 




830 


908 


913 


918 


923 


928 


934 


939 


944 


949 


955 




831 
832 
833 
834 
835 


960 
92 012 
064 
116 
168 


965 
017 
069 
122 
174 


970 
023 
075 
127 
179 


976 
028 
080 
132 
184 


981 
033 
085 
137 
189 


986 
038 
090 
142 
194 


99l 
043 
096 
148 
200 


996 
049 
101 
153 
205 


*002 
054 
106 
158 
210 


*007 
059 
111 
163 
215 




1 
2 
3 
4 
5 
6 
7 
8 
9 


5 

0.5 
1.0 
1-5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 


836 


220 


226 


231 


236 


241 


246 


252 


257 


262 


267 




837 
838 
839 


272 
324 
376 


277 
329 
381 


283 
335 
386 


288 
340 
391 


293 
345 
397 


298 
350 
4.02 


303 
355 
407 


309 
360 
412 


314 
366 
417 


319 
371 
423 




840 


428 


433 


438 


443 


448 


454 


459 


464 


469 


474 




841 


479 


485 


490 


495 


500 


505 


510 


515 


521 


526 




842 


531 


536 


541 


546 


552 


557 


562 


567 


572 


577 




843 


583 


588 


593 


598 


603 


608 


613 


619 


624 


629 




844 


634 


639 


644 


649 


655 


660 


665 


670 


675 


680 




845 


685 


691 


696 


701 


706 


711 


716 


721 


727 


732 




846 


737 


742 


747 


752 


757 


762 


768 


773 


778 


783 




847 


788 


793 


798 


803 


809 


814 


819 


824 


829 


834 




848 


839 


844 


850 


855 


860 


865 


870 


875 


880 


885 




849 


891 


896 


901 


906 


911 


916 


921 


926 


931 


937 




850 


942 


947 


952 


957 


962 


967 


972 
6 


977 

7 


982 
8 


988 




N. 





1 


3 


3 


4 


5 


9 


P. P. 



583 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 



850 

851 
852 
853 
854 
855 
856 
857 
858 
859 

860 

861 
862 
863 
864 
865 
866 
867 
868 
869 

870 

871 
872 
873 
874 
875 
876 
877 
878 
879 

880 

881 
882 
883 
884 
885 
886 
887 
888 
889 

890 

891 
892 
893 
894 
895 
896 
897 
898 



900 



N. 



92 942 947 




952 

94 002 
051 
101 
151 
201 
250 
300 
349 
399 

448 

497 
547 
596 
645 
694 
743 
792 
841 
890 

939 



95 036 
085 
134 
182 
231 
279 
327 
376 

424 



957 



007 
056 
106 
156 
206 
255 
305 
354 
404 



2 


3 








952 


957 


*003 


*008 


054 


059 


105 


110 


156 


161 


207 


212 


257 


262 


308 


313 


359 


364 


409 


414 


460 


465 


510 


515 


561 


566 


611 


616 


661 


666 


711 


716 


762 


767 


812 


817 


862 


867 


912 


917 


962 
012 


967 


017 


061 


0661 


111 


116, 


161 


1661 


210 


216: 



962 



^013 
064 
115 
1 

217 
267 
318 
369 
419 



470 

520 
571 
621 
671 
721 
772 
822 
872 
922 



972 



453 



502 
552 
601 
650 
699 
748 
797 
846 
895 



260 265 
310 315 
3591 364 
4091 413 



4581 463 



944 



992 
041 
090 
138 
187 
235 
284 
332 
381 



429 



507 
556 
606 
655 
704 
753 
802 
851 
900 

949 



512 
561 
611 
660 
709 
758 
807 
856 
905 



953 



997 
046 
095 
143 
192 
240 
289 
33 



^002 
051 
099 
148 
197 
245 
294 
342 



385; 390 



434 438 



022 
071 
121 
171 
220 
270 
320 
369 
418 



468 

517 
566 
615 
665 
714 
763 
812 
861 
909 



958 



'007 
056 
104 
153 
201 
250 
298 
347 
395 



443 



967i 972 
^0181*023 



069 
120 
171 



074 
125 
176 



222 227 
oao 278 
328 
379 
429 

475 480 



272 
323 
374 
424 



525 
576 
626 
676 
726 
777 
827 
877 
927 

977 



026 
076 
126 
176 
225 
275 
324 
374 
423 



473 

522 
571 
620 
670 
719 
768 
817 
865 
914 



963 

=012 
061 
109 
158 
206 
255 
303 
352 
400 



448 



530 
581 
631 
681 
731 
782 
832 
882 
932 



982 

03l 
081 
131 
181 
230 
280 
329 
379 
428 



478 



527 
576 
625 
674 
724 
773 
821 
870 
919 



968 

=017 
065 
114 
163 
211 
260 
308 
356 
405 

453 



977 



485 

535 
586 
636 
686 
736 
787 
837 
887 
937 



987 

036 
086 
136 
186 
235 
285 
334 
384 
433 



483 



532 
581 
630 
679 
728 
777 
826 
875 
924 



973 



=022 
070 
119 
167 
216 
264 
313 
361 
410 



458 



982 



^034 
084 
135 
186 
237 
288 
338 
389 
439 



490 

540 
591 
641 
691 
742 
792 
842 
892 
942 



992 



041 
091 

141 
191 
240 
290 
339 
389 
438 



487 

537 
586 
635 
684 
733 
782 
831 
880 
929 



978 

'"026 
075 
124 
172 
221 
269 
318 
366 
414 



463 



988 

^039 
090 
140 
191 
242 
293 
343 
394 
445 



495 



545 

596 

646 

69 

747 

797 

847 

897 

947 



997 



046 
096 
146 
196 
245 
295 
344 
394 
443 



492 

542 

591 

640 

689 

738 

787 

836 

88 

934 

983 

03l 
085 
129 
177 
226 
274 
323 
371 
419 

467 



P. P. 



.1 





.2 


1 


• 3 


1 


.4 


2 


• 5 


2 


• 6 


3 


•7 


3 


8 


4 


• 9 


4 



-1 





• 2 


1 


3 


1 


4 


2 


5 


2 


6 


3. 


7 


3- 


8 


4- 


9 


4. 



• 1 





.2 





■ 3 


1 


.4 


1 


.5 


2 


.6 


2. 


.7 


3 


.8 


3. 


.9 


4 



P.P. 



584: 







TABLE V 


—LOGARITHMS OF NUMBERS. 








N. 





1 


2 

434 


3 

438 


4 

443 


5 

448 


6 

453 


7 
458 


8 
463 


9 

467 


r. p. 


900 


95 424 


429 




901 


472 


477 


482 


487 


492 


496 


50l 


506 


511 


516 




902 


520 


525 


530 


535 


540 


544 


549 


554 


559 


564 




903 


569 


573 


578 


583 


588 


593 


597 


602 


607 


612 




904 


617 


621 


626 


631 


636 


641 


645 


65U 


6bb 


660 




905 


665 


669 


674 


679 


684 


689 


693 


698 


703 


708 




906 


713 


717 


722 


727 


732 


737 


741 


746 


751 


756 




907 


760 


765 


770 


775 


780 


784 


789 


794 


799 


804 




908 


808 


813 


818 


823 


827 


832 


837 


842 


847 


851 




909 


856 


861 


866 


870 


875 


880 


885 


890 


894 


899 




910 


904 


909 


913 


918 


923 


928 


933 


937 


942 


947 




911 


952 


956 


96l 


966 


971 


975 


980 


985 


990 


994 


1 


5 


912 


999 


*004 


*009 


*014 


*018 


*023 


*028 


*033 


*037 


*042 




2 
3 
4 
5 
6 
7 
8 
9 


1 
1 
2 
2 
3 
3 
4 
4 



5 

5 

5 

5 


913 


96 047 


052 


056 


061 


066 


071 


075 


080 


085 


090 




914 


094 


099 


104 


109 


113 


118 


123 


128 


132 


137 




915 


142 


147 


151 


156 


161 


166 


170 


175 


180 


185 




916 

917 


189 
237 


194 
241 


199 

246 


204 
251 


208 
256 


213 
260 


218 
265 


222 
270 


227 
275 


232 
279 




918 
919 


284 
331 


289 

336 


293 
341 


298 
345 


303 
350 


308 
355 


312 
360 


317 
364 


322 
369 


327 
374 




920 


379 


383 


388 


393 


397 


402 


407 


412 


416 


421 




921 


426 


430 


435 


440 


445 


449 


454 


459 


463 


468 




922 


473 


478 


482 


487 


492 


496 


501 


506 


511 


515 




923 


520 


525 


529 


534 


539 


543 


548 


553 


558 


562 




924 


567 


572 


576 


581 


586 


590 


595 


600 


605 


609 




925 


614 


619 


623 


628 


633 


637 


642 


647 


651 


656 




926 


661 


666 


670 


675 


680 


684 


689 


694 


698 


703 




927 


708 


712 


717 


722 


726 


731 


736 


741 


745 


750 




928 


755 


759 


764 


769 


773 


778 


783 


787 


792 


797 




929 


801 


806 


811 


815 


820 


825 


829 


834 


839 


843 




930 


848 


853 


857 


862 


867 


871 


876 


881 


885 


890 




931 
932 
933 


895 
941 
988 


899 
946 
993 


904 
951 
997 


909 

955 

=^002 


913 

960 

=^007 


918 

965 

*OlI 


923 

969 

*016 


927 

974 

*020 


932 

979 

*025 


937 

983 

*030 




1 
2 
3 
4 
5 
6 
7 
8 
9 





1 
1 
2 
2 
3 
3 
4 


\ 

9 
3 
8 
2 

I 

a 


934 


97 034 


039 


044 


048 


053 


058 


062 


067 


072 


076 




935 
936 


081 

127 


086 
132 


090 
137 


095 
141 


099 
146 


104 
151 


109 
155 


113 
160 


118 
164 


123 
169 




937 


174 


178 


183 


188 


192 


197 


202 


206 


211 


215 




938 
939 


220 
266 


225 
271 


229 
276 


234 
280 


239 
285 


243 
289 


248 
294 


252 
299 


257 
303 


262 
308 




940 


313 


317 


322 


326 


33l 


336 


340 


345 


349 


354 




941 


359 


363 


368 


373 


377 


382 


386 


39l 


396 


400 




942 


405 


409 


414 


419 


423 


428 


432 


437 


442 


446 




943 


451 


456 


460 


465 


469 


474 


479 


483 


488 


492 




944 


497 


502 


506 


511 


515 


520 


525 


529 


534 


538 




945 


543 


548 


552 


557 


561 


566 


570 


575 


580 


584 




946 


589 


593 


598 


603 


607 


612 


616 


621 


626 


630 




947 


635 


639 


644 


649 


653 


658 


662 


667 


671 


676 




948 


681 


685 


690 


694 


699 


703 


708 


713 


717 


722 




949 


726 


731 


736 


740 


745 


749 


754 


758 


763 


768 




950 


772 


777 
1 


781 
2 


786 


790 
4 


795 
5 


800 
6 


804 

7 


809 

8 


813 
9 




N. 





3 


p. p. 



585 







TABLE V.- 


-LOGARITHMS OF NUMBERS. 




N. 





1 


2 

781 


3 

786 


4 

790 


5 

795 


6 

800 


7 
804 


8 
809 


9 

813 


P. 


P. 


950 


97 772 


777 






951 


818 


822 


827 


83l 


836 


841 


845 


850 


854 


859 






952 


86S 


868 


873 


877 


882 


886 


891 


895 


900 


904 






953 


909 


914 


918 


923 


927 


932 


936 


941 


945 


950 






954 


955 


959 


964 


968 


973 


977 


982 


986 


991 


996 






955 


98 000 


005 


00^ 


014 


018 


023 


027 


032 


036 


041 






956 


046 


050 


055 


059 


064 


068 


073 


077 


082 


086 




6 ' 
0.5 
1.0 
1.5 


957 


091 


095 


100 


105 


109 


114 


118 


123 


127 


132 


.1 

.2 
.3 
.4 
.5 


958 
959 


136 
182 

227 


141 
186 

231 


145 
191 

236 


150 
195 

240 


154 
200 

245 


159 
204 

249 


163 
209 

254 


168 
213 

259 


173 
218 


177 
222 


960 


263 


268 


2.0 

2-5 


961 
962 
963 
964 


272 
317 
362 
407 


277 
322 
367 

412 


281 
326 
371 
416 


286 
331 
376 

421 


290 
335 
380 

425 


295 
340 
385 
430 


299 
344 
389 
434 


304 
349 
394 
439 


308 
353 
398 
443 


313 
358 
403 
448 


.6 
.7 
.8 
•9 


30 
3.5 
4.0 

4.9 


965 


452 


457 


461 


466 


470 


475 


479 


484 


488 


493 






966 


497 


502 


506 


511 


515 


520 


524 


529 


533 


538 






967 


542 


547 


551 


556 


560 


565 


569 


574 


578 


583 






968 


587 


592 


596 


601 


605 


610 


614 


619 


623 


62{ 






969 


632 


637 


641 


646 


650 


655 


659 


663 


668 


672 






970 


677 


681 


686 


690 


695 


699 


704 


70§ 


713 


717 






971 


722 


726 


731 


735 


740 


744 


749 


753 


757 


762 


1 

.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


0^ 
0.9 
1-3 

2^2 

U 

u 


972 
973 


766 
811 


771 
815 


775 
820 


780 
824 


784 
829 


789 
833 


793 
838 


798 
842 


802 
847 


807 
85l 


974 
975 
976 


856 
900 
945 


860 
905 
949 


865 
909 
954 


869 
914 
958 


873 
918 
963 


878 

922 
967 


882 
927 
971 


887 
931 
976 


891 
936 
980 


896 
940 
985 


977 


989 


994 


998 


*003 


*007 


*011 


*016 


*020 


*025 


*029 


978 


99 034 


038 


043 


0^7 


051 


056 


060 


065 


069 


074 


979 


078 
122 
167 


082 
127 
171 


087 
131 
176 


091 
136 
180 


096 
140 
184 


100 
145 
189 


105 
149 
193 


109 
153 
198 


113 
158 


118 


980 


162 


981 


202 


206 


982 


21 


215 


220 


224 


229 


23o 


237 


242 


246 


251 






983 


255 


260 


264 


268 


273 


277 


282 


286 


290 


295 






984 


299 


304 


308 


312 


317 


32: 


326 


330 


335 


339 






985 


343 


348 


352 


357 


361 


365 


370 


374 


379 


383 






986 


387 


392 


396 


401 


405 


409 


414 


418 


423 


427 




4 

0.4 
0.8 

1.2 


987 


431 


436 


440 


445 


449 


45^ 


458 


462 


467 


471 


.1 

.2 
• 3 


988 


475 


480 


484 


489 


493 


497 


502 


506 


511 


515 


989 


519 


524 


528 


533 


537 


541 


546 


550 


554 


559 


990 


563 


568 


572 


576 


581 


585 


590 


594 


598 


603 


.4 
• 5 
.6 
.7 
.8 
.9 


1.6 
2.0 
2.4 
2.8 
3.2 
3.6 


991 


607 


6ll 


616 


620 


625 


629 


633 


638 


642 


647 


992 


651 


655 


660 


664 


668 


673 


677 


682 


686 


690 


993 


695 


699 


703 


708 


712 


717 


721 


725 


730 


734 


994 


738 


743 


747 


751 


756 


760 


765 


769 


773 


778 


995 


782 


786 


791 


795 


800 


804 


808 


813 


817 


821 






996 


826 


830 


834 


839 


843 


847 


852 


856 


861 


865 






997 


869 


874 


878 


882 


887 


891 


895 


900 


904 


908 






998 


913 


917 


922 


926 


930 


935 


939 


943 


948 


952 






999 


956 


961 


965 


969 


974 


978 


982 


987 


991 


995 






1000 


00 000 


004 


008 
2 


013 
3 


017 
4 


021 
5 


026 
6 


030 
7 


034 
8 


039 
9 


N. 





1 


P. 1 


P. 






586 









































TABLE V - 


-LOGARITHMS OP 


^ NUMBERS. • 










N. 





1 
043 


2 

087 


3 

130 


4 

173 


5 

217 


6 

260 


7 
304 


8 

347 


9 

390 


P. P. 


1000 


000 000 




01 


434 


477 


521 


564 


607 


651 


694 


737 


781 


824 




02 


867 


911 


954 


997 


*041 


*084 


*127 


^171 


*214 


^257 




03 


001 301 


344 


387 


431 


474 


517 


560 


604 


647 


690 




04 


733 


777 


820 


863 


906 


950 


993 


*036*079 


*123 




05 


002 166 


209 


252 


295 


339 


382 


425 


468 511 


555 




06 


598 


641 


684 


727 


770 


814 


857 


900 943 


986 


/lO /I o 


07 


003 029 


072 


115 


159 


202 


245 


288 


33l 374 


417 


.1 

.2 
.3 


A 5 


A. ^ 


08 


460 


503 


546 


590 


633 


676 


719 


762i 805 


848 


8. 

13. 


7 



8. 

12. 


6 
9 


09 


891 


934 


977 


*020 


*063 


no6 


*149 


n92 


*235 


*278 


1010 


004 321 
751 


364 
794 


407 
837 


450 
880 


493 
923 


536 
966 


579 
*009 


622 
*05l 


665 
*094 


708 


.4 
.5 
.6 
•7 
.8 
• 9 


17. 
21- 
26 
30 
34 

00 


4 
7 

1 
4 

-? 


17. 
21. 
25. 
30. 
34. 
38. 


2 
5 


11 


n37 


8 

1 
4 

7 


12 


005 180 


223 


266 


309 


352 


395 


438 


481 


523 


566 


13 


609 


652 


695 


738 


781 


824 


866 


9091 952 


995 


14 


006 038 


081 


123 


166 


209 


252 


295 


337 


380 


423 






15 


466 


509 


551 


594 


637 


680 


722 


765 


808 


851 




16 


893 


936 


979 


*022 


*06^: 


*107 


*150 


*193 


*235 


*278 




17 


007 321 


363 


406 


449 


49: 


534 


577 


620 


662 


705 




18 


748 


790 


833 


875 


91 


961 


*003 


*046 


*089 


131 




19 


008 174 
600 


217 
642 


259 
685 


302 
728 


344 
770 


387 


430 


472 


515 


557 




1030 


813 


855 


898 


940 


983 




21 


009 025 


068 


111 


153 


196 


238 


281 


323 


366 


408 


.1 
.2 
.3 
.4 
.5 
.6 
-7 
.8 
.9 


42 

4 ^ 


42 

4 ^ 


22 


451 


493 


536 


578 


621 


663 


706 


748 


790 


833 


8 

12 
17 
21 
25 
29 
34 
38 


5 
7 

5 
7 


8 

12 
16 
21 
25 
29 
33 
37 


4 
6 
8 

2 
4 
6 
8 


23 


875 


918 


960 


*003 


*045 


*088 


^130 


*172 *215 


*257 


24 


010 300 


342 


385 


427 


469 


512 554 


596 639 


681 


25 


724 


766 


808 


851 


893 


935 


978 


*020i*062 


*105 


26 


Oil 147 


189 


232 


274 


316 


359 


401 


4431 486 


528 


27 


570 


612 


655 


697 


739 


782 


824 


866 908 


951 


28 


993 


*035 


*077 


'*=120 


*162 


*204 


*246 


*288 *331 


*373 


29 


012 415 


457 


500 


542 


584 


626 


668 
*090 


710 753 
*132 174 


795 
216 


1030 


837 


879 


92l 


963 


*0t6 


*048 




31 


013 258 


301 


343 


385 


427 


469 


511 


553 595 


637 




32 


679 


722 


764 


806 


848 


890 


932 


974*016 


*058 




33 


014 100 


142 


184 


226 


268 


310 


352 


394 436 


478 




34 


520 


562 


604 


646 


688 


730 


772 


814 856 


898 




35 


940 


982 


*024 


*066 


*108 


*150 


*192 


*234*276 


*318 




36 


015 360 


401 


443 


485 


527 


569 


611 


653: 695 


737 


/iT A-i 


37 


779 


820 


862 


904 


946 


988 


*030 


*072*113 


155 


.1 
.2 
.3 
-4 
.5 
.6 
.7 
.8 
o 


1 ^ 


4 1 


38 


016 197 


239 


281 


323 


364 


406 


448 


490 532 


573 


8 

12 
16 
20 
24 
29 
33 


•3 
4 
•6 
•7 

i 

.2 
5 


8 

12 
16 
20 
24 
28 
32 


• 2 
3 


39 


615 
017 033 


657 
075 


699 
117 


741 
158 


782 
200 


824 
242 


866 
284 


908 950 
325 367 


991 


1040 


409 


• 4 
5 


41 


450 


492 


534 


576 


617 


659 


701 


742 784 


826 


6 
■7 


42 


867 


909 


951 


992 


*034 


*076 


*117 


*159 *201 


*242 


43 


018 284 


326 


367 


409 


451 


492 


534 


575 617 


659 


QR.Q 


44 


700 


742 


783 


825 


867 


908 


950 


99l *033 


*074 




45 


019 116 


158 


199 


241 


282 


324 


365 


407 448 


490 




46 


531 


573 


614 


656 


697 


739 


780 


822 863 


905 




47 


946 


988 


*029 


*071 


*n2 


*154 


*195 


*237*278 


*320 




48 


020 36l 


402 


444 


485 


527 


568 


610 


65l 692 


734 




49 


775 
021 189 


817 
230 


858 
272 


899 
313 


941 
354 


982 
396 


*024 


*065*106 
478 520 


*148 
56l 




1050 


437 




N. 





1 


2 


3 


4 


5 


6 


7 


« 


9 


P.P. 



587 









TABLE V.- 


-LOGARITHMS OF NUMBERS. 






1 


N. 





1 


2 


3 


4 


5 


6 


7 
478 


8 
520 


9 


P 


.P. 


1050 


021 


189 


230 


272 


313 


354 


396 


437 


56l 




i 

41 

4.1 ! 


51 




602 


644 


685 


726 


768 


809 


850 


892 


933 


974 


1 


52 


022 


015 


057 


098 


139 


181 


222 


263 


304 


346 


387 


2 


8 


3 


53 




428 


469 


511 


552 


593 


634 


676 


717 


758 


799 


3 


12 


4 


54 




840 


882 


923 


964 


-^005 


*046 


'^088 


-^129 


^•'170 


*211 


4 


16 


6 


55 


023 


252 


293 


335 


376 


417 


458 


499 


540 


581 


623 


5 


20 


7 


56 




664 


705 


746 


787 


828 


869 


910 


951 


993 


*034 


6 


24 


9 


57 


024 


075 


116 


157 


198 


239 


280 


321 


362 


403 


444 


7 


29 





58 




485 


526 


568 


609 


650 


691 


732 


773 


814 


855 


8 


33 


2 


59 


025 


896 


937 


978 


*019 


*060 


^101 


■^142 


*183 


*224 


*265 


9 

1 


37 


3 


1060 


306 


347 


388 


429 


469 


510 


551 


592 


633 


674 


41 

41 


61 


715 


756 


797 


838 


879 


920 


961 


*002 


*042 


*085 


62 


026 


124 


165 


206 


247 


288 


329 


370 


410 


451 


492 


2 


8 


2 


63 




533 


574 


615 


656 


696 


737 


778 


819 


860 


901 


3 


12 


3 


64 




941 


982 


*023 


^064 


^105 


n45 


^^186 


^■227 


=^268 


*309 


4 


16. 


4 
5 


65 


027 


349 


390 


431 


472 


512 


553 


594 


635 


675 


716 


5 


20 


66 




757 


798 


838 


879 


920 


961 


*001 


*042 


*083 


n23 


6 


24 


6 


67 


028 


164 


205 


246 


286 


327 


368 


408 


449 


490 


530 


7 


28 


7 


68 




571 


612 


652 


693 


734 


774 


815 


856 


896 


937 


8 


32 


8 


69 


029 


977 
384 


*018 
424 


*059 
465 


=*=099 


*140 


*181 


=223 
627 


*262 
668 


*302 
708 


*343 
749 


9 


36 


9 


1070 


505 


546 


586 


40 1 

4.0 1 


71 




789 


830 


870 


911 


951 


992 


*032 


*073 


*114 


*154 


1 


72 


030 


195 


235 


276 


316 


357 


397 


438 


478 


519 


559 


2 


8 


1 


73 




599 


640 


680 


721 


761 


802 


842 


883 


923 


964 


3 


12 


1 


74 


031 


004 


044 


085 


125 


166 


206 


247 


287 


327 


368 


4 


16 


2 


75 




408 


449 


489 


529 


570 


610 


651 


691 


731 


772 


5 


20 


2 


76 




812 


852 


893 


933 


973 


^=^014 


=^054 


-^094 


^135 


n75 


6 


24 


3 


77 


032 


215 


256 


296 


336 


377 


417 


457 


498 


538 


578 


7 


28 


3 


78 




619 


659 


699 


739 


780 


820 


860 


900 


941 


981 


8 


32 


i . 


79 


033 


021 


061 


102 


142 


182 


222 


263 


303 


343 


383 


9 


36 


4 


1080 




424 


464 


504 


544 


584 


625 
*026 


665 
*066 


705 
*107 


745 
147 


785 
187 


1 


40 

4 


81 


825 


866 


906 


946 


986 


82 


034 


227 


267 


307 


347 


388 


428 


468 


508 


548 


588 


2 


8 





83 




628 


668 


708 


748 


789 


829 


869 


909 


949 


989 


3 


12 





84 


035 


029 


069 


109 


149 


189 


229 


269 


309 


349 


389 


4 


16 





85 




429 


470 


510 


550 


590 


630 


670 


710 


750 


790 


5 


20 





86 




830 


870 


9101 950 


990 


=^029 


=^069 


*109 


*149 


*189 


6 


24 





87 


036 


229 


269 


309i 349 


389 


429 


469 


509 


549 


589 


7 


28 





88 




629 


669 


708! 748 


788 


828 


868 


908 


948 


988 


8 


32 





89 


037 


028 


068 


107 147 


187 


227 


267 


307 


347 


386 


9 


36 





1090 




426 


466 


506 


546 


586 


625 


665 


705 


745 


785 






91 




825 


864 


904 


944 


984 


*023 


*063 


*103 


143 


183 


• 1 


•5 

3 


•g 


92 


038 


222 


262 


302 


342 


381 


421 


461 


501 


540 


580 


• 2 


7 




93 




620 


660 


699 


739 


779 


819 


858 


898 


938 


977 


• 3 


11 


8 


94 


039 


017 


057 


096 


136 


176 


216 


255 


295 


335 


374 


• 4 


15 


8 ^ 


95 




414 


454 


493 


533 


572 


612 


652 


691 


731 


771 


.5 


19 


•7 


96 




810 


850 


890 


929 


969 


=^008 


===048 


*088 


*127l*167 


.6 


23 


• 7 


97 


040 


206 


246 


286 


325 


365 


404 


444 


483 


523 563 


• 7 


27 


• 6 


98 




602 


642 


681 


721 


760 


800 


839 


879 


918 958 


.8 


31 


• 6 


99 


041 


997 
392 


*037 
432 


*076 
471 


*116 
511 

3 


n55 
550 

4 


=^195 
590 

5 


^234 
629 

6 


*274 
669 

7 


*313j*353 


.9 


35 


■5 


1100 


708 


748 




N. 





1 


2 


8 


9 


F 


.P. 



588 



TABLE VI.— 


LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 


Log: sin = log ^'' + S. 




0° 


log 4>' 


' = log sin 


+ 8'. 


Log tan cfy = log </>'' + T. 




. log 0' 


' = log tan + T', 


1/ 


t 


S 


T \.i 


3g. Sin. 


S' 


T' I.og. Tan. 








4-685 57 


57 


CO 


5.314 42 


42 


00 


60 ^ 


1 


57 


57 6 


.46 372 


42 


42 6 


-46 372 


120 * 


2 


57 


57 


.76 475 


42 


42 


• 76 475 


180 


3 


57 


57 


• 94 084 


42 


42 


-94 084 


240 


4 


57 


57 7 


.06 578 


42 
5.31442 


42 7 
42 ■ 7 


• 06 578 


300 


5 


4-685 57 


57 7 


-16 269 


-16 269 


360 


6 


57 


57 


-24 187 


42 


42 


24 188 


420 


7 


57 


57 


-30 882 


42 


42 


-30 882 


480 


8 


57 


57 


36 681 


42 


42 


-36 681 


540 


9 


57 


57 


.41 797 


42 


42 


41 797 


600 


10 


4-685 57 


57 7 


• 46 372 


5.31442 


42 : 7 


■ 46 372 


660 


11 


57 


57 


.50 512 


42 


42 


-50 512 


720 


12 


57 


57 


-54 290 


42 


42 


-54 291 


780 


13 


57 


57 


-57 767 


42 


42 


-57 767 


840 


14 


57 


57 


-60 985 


42 


42 


■ 60 985 


900 


15 


4. 685 57 


58 7 


-63 981 


5.314 42 


42 7 


-63 982 


960 


16 


57 


58 


-66 784 


42 


42 


-66 785 


1020 


17 


57 


58 


-69 417 




42 


.69 418 


1080 


18 


57 


58 


-71 899 


42 


42 


• 71 900 


1140 


19 


57 


58 


-74 248 


42 


42 


- 74 248 


1200 


20 


4-685 57 


58 7 


-76 475 


5.314 43 


42 7 


76 476 


1260 


21 


57 


58 


-78 594 


43 


42 


.78 595 


1320 


22 


57 


58 


-80 614 


43 


42 


-80 615 


1380 


23 


57 


58 


.82 545 


43 


42 


82 546 


1440 


24 


57 


58 


84 393 


43 


42 


84 394 


1500 


25 


4685 57 


58 7 


86 166 


5-31443 


41 7 


86 167 


1560 


26 


57 


58 


87 869 


43 


41 


87 871 


1620 27 


57 


5g 


89 508 


43 


4 


89 510 


1680 


28 


57 


58 


91 088 


43 


41 


91089 


1740 


29 


57 


58 


92 612 


43 


4 


92 613 


1800 


30 


4-685 57 


58 7 


94 084 


5-314 43 


41 i 7 


94 086 


1860 


31 


57 


58 


95 508 


43 


4 


95 510 


1920 


32 


57 


58 


96 887 


43 


4 


96 889 


1980 


33 


57 


59 


98 223 


43 


41 


98 225 


2040 


34 


57 


59 


99 520 


43 


41 


99 522 


2100 


35 


4.685 56 


59 8 


00 778 


5.31443 


41 8 


00 781 


2160 


36 


56 


59 


02 002 


43 


41 


02 004 


2220 


37 


56 


59 


03 192 


43 


41 


03 194 


2280 


38 


56 


59 


04 350 


43 


40 


04 352 


2340 


39 


56 


59 


05 478 


43 


40 


05 481 


2400 


40 


4. 685 56 


59 8- 


06 577 


5.31443 


40 8- 


06 580 


2460 


41 


56 


59 


07 650 


43 


40 • 


07 653 


2520 


42 


56 


59 


08 696 


43 


40 


08 699 


2580 


43 


56 


60 


09 718 


43 


40 


09 721 


2640 


44 


56 


60 


10 716 


43 


40 


10 720 


2700 


45 


4-685 56 


60 8. 


11 692 


5.314 44 


40 8. 


11 696 


2760 


46 


56 


60 


12 647 


44 


40 


12 651 


2820 


47 


56 


60 


13 581 


44 


40 


13 585 


2880 


48 


56 


60 


14 495 


44 


39 


14 499 


2940 
3000 


49 


56 


60 


15 390 


44 


39 


15 395 


50 


4-685 56 


60 8- 


16 268 


5-314 44 


39 8- 


16 272 


3060 


51 


56 


60 


17 128 


44 


39 


17 133 


U20 


52 


56 


61 


17 971 


44 


39 


17 976 


3180 


53 


56 


61 


18 798 


44 


39 


18 803 


3240 
3300 


54 
55 


55 


61 


19 610 


44 


39 


19 615 


4-685 55 


61 8 


20 407 


5-314 44 


39 8- 


20 412 


3360 


56 


55 


61 


21 189 


44 


38 


21 195 


3420 


57 


55 


61 


21 958 


44 


38 


21 964 


3480 


58 


55 


61 


22 713 


44 


38 


22 719 


3540 


59 


55 


62 


23 455 


44 


38 


23 462 



1 



589 



TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 



Log sin <f> = log (f)'' 4- S. 




r 


log </)'' 


= log sin ^ + /S^ 


Log taa (f> = log gS" 4- T. 




log 0" 


= log tan <^ -^ T' . 


" 


# 


S 


T \.i 


3g. Sin. 


S' 


T' I.og. Tan. 


3600 





4.685 55 


62 8 


-24 185 


5-314 44 


38 8 


• 24 192 


3660 


1 


55 


62 


-24 903 


45 


38 


-24 910 


3720 


2 


55 


62 


-25 609 


45 


38 


• 25 616 


3780 


3 


55 


62 


.26 304 


45 


37 


• 26 311 


3840 


4 


55 


62 


26 988 


45 


37 1 


• 26 995 


3900 


5 


4-685 55 


62 8 


• 27 661 


5-31445 


37 8 


27 669 


3960 


6 


55 


63 


-28 324 


45 


37 


• 28 332 


4020 


7 


54 


63 


-28 977 


45 


37 


• 28 985 


4080 


8 


54 


63 


-29 620 


45 


37 ; 


•29 629 


4140 


9 


54 


63 


30 254 


45 
5-31445 


36 1 


• 30 263 


4200 


10 


4-685 54 


63 8 


30 879 


36 8 


• 30 888 


4260 


11 


54 


63 


31 495 


45 


36 


•31 504 


4320 


12 


54 


64 


32 102 


45 


36 


-32 112 


4380 


13 


54 


64 


32 701 


46 


36 


• 32 711 


4440 


14 


54 


64 


33 292 


46 


36 


• 33 302 


4500 


15 


4-685 54 


64 8 


33 875 


5-31446 


35 8 


33 885 


4560 


16 


54 


64 


34 450 


46 


35 


-34 461 


4620 


17 


54 


65 


35 018 


46 


35 


-35 029 


4680 


18 


54 


65 


35 578 


46 


35 


• 35 589 


4740 


19 


53 


65 


36 131 


46 


35 


•36 143 


4800 


20 


4. 685 53 


65 8 


36 677 


5-314 46 


34 8 


36 689 


4860 


21 


53 


65 


37 217 


46 


34 


-37 229 


4920 


22 


53 


65 


37 750 


46 


34 


37 762 


4980 


23 


53 


66 


38 276 


46 


34 


38 289 


5040 


24 


53 


66 


38 796 


47 
5-314 47 


34 


38 809 


5100 


25 


4-685 53 


66 8 


39 310 


33 8 


39 323 


5160 


26 


53 


66 


39 818 


47 


33 


39 831 


5220 


27 


53 


67 


40 320 


47 


33 


40 334 


5280 


28 


52 


67 


40 816 


47 


33 


40 830 


5340 


29 


52 


67 


41 307 


47 


33 


41 321 


5400 


30 


4-685 52 


67 8 


41 792 


5-31447 


32 8 


41 807 


5460 


31 


52 


67 


42 271 


47 


32 


42 287 


5520 


32 


52 


68 


42 746 


47 


32 


42 762 


5580 


33 


52 


68 


43 215 


48 


32 


43 231 


5640 


34 


52 


68 


43 680 


48 


31 


43 696 


5700 


35 


4-685 52 


68 8 


•44 139 


5-31448 


31 8 


44 156 


5760 


36 


52 


69 


44 594 


48 


31 


44 611 


5820 


37 


5: 


69 


45 044 


48 


31 


45 061 


5880 


38 


5:; 


69 


45 489 


48 


30 


45 507 


5940 


39 


5l 


69 


45 930 


48 
5-314 48 


30 


45 948 


6000 


40 


4-685 51 


69 8 


46 366 


30 8 


46 385 


6060 


41 


51 


70 


46 798 


49 


30 


46 817. 


6120 


42 


51 


70 


47 226 


49 


30 


47 245 


6180 


43 


51 


70 


47 650 


49 


29 


47 669 


6240 


44 


51 


70 


48 069 


49 


29 


48 089 


6300 


45 


4-685 50 


71 8 


48 485 


5-31449 


29 8 


48 505 


6360 


46 


50 


71 


48 896 


49 


28 


48 917 


6420 


47 


50 


71 


49 304 


49 


28 


49 325 


6480 


48 


50 


72 


49 708 


49 


28 


49 729 


6540 


49 


50 


72 


50 108 


50 


28 


50 130 


6600 


50 


4-685 50 


72 8 


50 504 


5-314 50 


27 8 


50 526 


6660 


51 


50 


72 


50 897 


50 


27 


50 920i 


6720 


52 


50 


73 


51 286 


50 


27 


51 310: 


6780 


53 


49 


73 


51 672 


50 


27 


51 696* 


6840 


54 


49 


73 


52 055 


50 


26 


11 079' 
52 458 


6900 


55 


4.685 49 


73 8 


52 434 


5-314 50 


26 8- 


6960 


56 


49 


74 


52 810 


51 


26 


52 835 


7020 


57 


49 


74 ■ 


53 183 


51 


25 


53 208 


7080 


58 


49 


74 


53 552 


51 


25 


53 571 


7140 


59 


49 


75 


53 918 


51 


25 


53 944 








r 


m 









TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES 



Log sin ^ = log (f)" + S. 




2° 


log 0' 


' = log sin 4> + S'. 


Log tan <f> = log 0'' + T. 




log0' 


= log tan <f) •¥ T', 


ff 


/ 


S 


T Log. Sin. 


S' 


T' Log 


^ Tan. 


7200 





4. 685 48 


75 8 


.54 282 


5-314 51 


25 8 


54 308 


7260 


1 


48 


75 


.54 642 


51 


24 


54 669 


7320 


2 


48 


75 


.54 999 


5 . 


24 


55 027 


7380 


3 


48 


76 


55 354 


52 


24 


55 381 


7440 


4 


48 


76 


55 705 


52 


23 


55 733 


7500 


5 


4-685 48 


76 8 


56 054 


5.314 52 


23 8 


56 083 


7560 


6 


48 


77 


56 400 


52 


23 


56 429 


7620 


7 


47 


77 


56 743 


52 


22 


56 772 


7680 


8 


47 


77 


57 083 


52 


22 


57 113 


7740 


9 


47 


78 


57 421 


52 


22 


57 452 


7800 


10 


4. 685 47 


78 8 


57 756 


5.314 53 


22 8 


57 787 


7860 


11 


47 


78 


58 089 


53 


21 


58 121 


7920 


12 


47 


79 


58 419 


53 


21 


58 451 


7980 


13 


46 


79 


58 747 


53 


21 


58 779 


8040 


14 


46 


79 


59 072 


53 


20 


59 105 


8100 


15 


4.685 46 


80 8 


59 395 


5-314 53 


20 8 


59 428 


8160 


16 


46 


80 


59 715 


54 


20 


59 749 


8220 


17 


46 


80 


60 033 


54 


19 


60 067 


8280 


18 


46 


81 


60 349 


54 


19 


60 384 


8340 


19 


45 


81 


60 662 


54 


19 


60 698 


8400 


20 


4.685 45 


8l 8 


60 973 


5-314 54 


18 8 


61 009 


8460 


21 


45 


82 


61 282 


54 


18 


61 319 


8520 


22 


45 


82 


61 589 


55 


18 


61 626 


8580 


23 


45 


82 


61 893 


55 


17 


61 931 


8640 


24 


45 


83 


62 196 


55 


17 


62 234 


8700 


25 


4.685 44 


83 8 


62 496 


5-314 55 


16 8 


62 535 


8760 


26 


44 


83 


62 795 


55 


16 


62 834 


8820 


27 


44 


84 


63 091 


55 


16 


63 13] 


8&80 


28 


44 


84 


63 385 


56 




63 425 


89'4'0 


29 


44 


84 


63 677 


56 


15 


63 718 


9000 


30 


4.685 43 


85 8 


63 968 


5.314 56 


15 8 


64 009 


9060 


31 


43 


85 


64 256 


56 


14 


64 298 


9120 


32 


43 


86 


64 543 


56 


14 


64 585 


9180 


33 


43 


86 


64 827 


57 


14 


64 870 


9240 


34 


43 


86 


65 110 


57 


13 


65 153 


9300 


35 


4.685 43 


87 8 


65 391 


5.314 57 


13 8 


65 435 


9360 


36 


42 


87 


65 670 


57 


12 


65 715 


9420 


37 


42 


87 


65 947 


57 


12 


65 993 


9480 


38 


42 


88 


66 223 


58 


12 


66 269 


9540 


39 


42 


88 


66 497 


58 


11 


66 543 


9600 


40 


4.685 42 


89 8 


66 769 


5.314 58 


11 8 


66 816 


9660 


41 


4;: 


89 


67 039 


58 


10 


67 087 


9720 


42 


4 


89 


67 308 


58 


10 


67 356 


9780 


43 


41 


90 


67 575 


59 


10 


67 624 


9840 


44 


41 


90 


67 840 


59 


09 


67 890 


9900 


45 


4.685 41 


91 8. 


68 104 


5-314 59 


09 8 


68 154 


9960 


46 


40 


91 


68 36G 


59 


08 


68 417 


10020 


47 


40 


91 


68 627 


59 


08 


68 678 


10080 


48 


40 


92 


68 886 


60 


08 


68 938 


10140 


49 


40 


92 


69 144 


60 


07 


69 196 


10200 


50 


4.685 40 


93 8 


69 400 


5.314 60 


07 8 


69 453 


10260 


51 


39 


93 


69 654 


60 


06 


69 708 


10320 


52 


39 


93 


69 907 


60 


06 


69 961 


10380 


53 


39 


94 


70 159 


61 


06 


70 214 


10440 


54 
55 


39 


94 


70 409 


61 


05 


70 464 


10500 


4. 685 38 


95 8 


70 657 


5.31461 


05 8 


.70 714 


10560 


56 


38 


95 


70 905 


61 


04 


.70 962 


10620 


57 


38 


96 


71 150 


61 


04 


.71 208 


10680 


58 


38 


96 


71 395 


62 


03 


.71 453 


10740 


59 


38 


97 


71 638 


62 


03 


.71 697 



591 



TABLE VII. 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



179° 



' 


Log. Sin. 


D 


Log, Tan. 


Com. D. 


Log. Cot. 


Log. Cos, 






1 

2 
3 
4 


6 
6 
6 
7 


46 372 
76 475 
94 084 
06 578 


30103 
17609 
12494 

9691 
7918 
6695 
5799 
5115 
4575 
4139 
3778 
3476 
3218 

2996 
2803 
2633 
2482 
2348 

2227 
2119 
2020 
1930 
1848 
1772 
1703 
1639 
1579 
1524 
1472 
1424 
1379 
1336 
1296 
1258 
1223 
1190 
1158 
1128 
1099 
1072 
1046 
1022 
998 
976 
954 
934 
914 
895 

877 
860 
843 
827 
8ll 
797 
782 
768 
755 
742 
730 


— O) 

6. 46 372 
6-76 475 
6. 94 084 
7. 06 578 


30103 
17609 
12494 

9691 
7918 
6694 
5799 
5115 

4575 
4139 
3779 
3476 
3218 
2996 
2803 
2633 
2482 
2348 
2227 
2119 
2020 
1930 
1848 

1773 
1703 
163£ 
1579 
1524 
1472 
1424 
1379 
1336 
1296 
1259 
1223 
1190 
1158 
1128 
1099 
1072 
1046 
1022 
999 

976 
954 
934 
914 
895 
877 
860 
843 
827 
812 
797 
783 
768 
755 
742 
730 


3 
3 
3 

2 


4-00 

53 627 
23 524 
05 915 
93 421 









00 000 
00 000 
00 000 

• 00 000 

• 00 000 


60 

59 
58 
57 
5b 


5 
6 
7 
8 
9 


7 
7 
7 
7 
7 


16 269 
24 187 
30 882 
36 681 
41 797 


7. 16 269 
7. 24 188 
7. 30 882 
7-36 681 
7-41 797 


2 
2 
2 
2 
2 


83 730 
• 75 812 
69 117 
63 318 
58 203 









• 00 000 

• 00 000 

• 00 000 

• 00 000 

• 00 000 


55 
54 
53 
52 
51 


10 

11 

12 
13 
14 


7 

7 
7 
7 
7 


46 372 
50 512 
54 290 
57 767 
60 985 


7-46 372 
7-50 512 
7-54 291 
7-57 767 
7. 60 985 

7. 63 982 
7. 66 785 
7-69 418 
7-71 900 
7. 74 248 


2 
2 
2 
2 
2 


• 53 627 
49 488 

.45 709 
42 233 
39 014 




9 
9 
9 


• 00 000 
00 000 

• 99 999 
99 999 
99 999 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


7 
7 
7 
7 
7 


63 981 
66 784 
69 417 
71 899 
74 248 


2 

2 
2 
2 
2 


• 36 018 
■ 33 215 
30 582 
28 099 

25 751 


9 
9 
9 
9 
9 


99 999 
99 999 
99 999 
• 99 999 
99 999 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


7 
7 
7 
7 
7 


76 475 
78 594 
80 614 
82 545 
84 393 


7. 76 476 
7-78 595 
7. 80 615 
7. 82 546 
7. 84 394 

7 86 167 
7-87 871 
789 510 
7 91 089 
7.92 613 


2 
2 

2 
2 
2 


23 524 
• 21 405 

■ 19 384 

■ 17 454 
15 605 


9 
9 
9 
9 
9 


• 99 999 
99 999 
99 999 
99 999 

■ 99 999 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


7 
7 
7 
7 
7 


86 166 

87 869 
89 508 

91 088 

92 612 


2 
2 
2 

I 


■ 13 832 

■ 12 129 

■ 10 490 

■ 08 910 

■ 07 386 


9 
9 
9 
9 
9 


■ 99 999 
99 999 

■ 99 998 
99 998 
99 998 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


7 
7 
7 
7 
7 


94 084 

95 508 

96 887 

98 223 

99 520 


7 94 086 

7. 95 510 

7. 96 889 
7. 98 225 
7-99 522 
8. 00 781 
8. 02 004 
803 194 

8 04 352 
8 05 481 


2 
2 
2 
2 
2 


05 914 
04 490 
03 111 
01 774 
■ 00 478 


9 

9 
9 
9 
9 


99 998 
99 998 
99 998 
99 998 
99 998 


30 

29 
28 

27 
26 


35 
36 
37 
38 
39 


8 
8 
8 
8 
8 


00 778 

02 002 

03 192 

04 350 

05 478 




99 219 
97 995 
96 805 
95 647 
94 519 


9 
9 
9 
9 
9 


99 997 
99 997 
99 997 
99 997 
99 997 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


8 
8 
8 
8 
8 


06 577 

07 650 

08 696 

09 718 

10 716 


8. 06 580 
8 07 653 
8 08 699 

8. 09 721 

8. 10 720 




93 419 
92 347 
91 300 
90 278 
89 279 


9 
9 
9 
9 
9 


99 997 
99 997 
99 997 
99 996 
99 996 


20 

19 
18 

17 
16 


45 
46 
47 
48 
49 


8 
8 
8 
8 
8 


11 692 

12 647 

13 581 

14 495 

15 390 


8. 11 696 
8-12 651 

8. 13 585 

8. 14 499 
815 395 




88 303 
87 349 
86 415 
85 500 
84 605 


9 
9 
9 
9 
9 


99 996 
99 996 
99 996 
99 996 
99 995 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


8 
8 
8 
8 
8 


16 268 

17 128 

17 971 

18 798 

19 610 


8-16 272 
8-17 133 
817 976 
8 18 803 
819 615 




83 727 
82 867 
82 023 
81 196 
80 384 


9 
9 
9 
9 
9 


99 995 
99 995 
99 995 
99 995 
99 994 


10 

9 
8 

7, 
6f 


55 
56 
57 
58 
59 


8 
8 
8 
8 
8 


20 407 

21 189 

21 958 

22 713 

23 455 


8-20 412 
8. 21 195 
8-21 964 
8-22 719 
8-23 462 




79 587 
78 804 
78 036 
77 280 
76 538 


9 
9 
9 
9 
9 


99 994 
99 994 
99 994 
99 994 
99 993 


5^ 
4 
3 

2 

1 


60 


8 


24 185 


8. 24 192 




75 808 


9 


99 993 


O 




Log. Cos. 


D 


Log. Cot. 


Com. D. 


Log. Tan. j 


Log. Sin. 


' 



90° 



592 



89*^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS 
AND COTANGENTS. 




TABLE VII. 



-LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



177° 



Log. Sin, 



D 



8-54 282 
8.54 642 
8.54 999 
8. 55 354 
8-55 705 



8.56 054 
8.56 400 

8.56 743 

8.57 083 
8 57 421 



8.57 756 

8.58 039 
8.58 419 

8.58 747 

8.59 072 



8.59 395 
8-59 715 

8.60 033 
8.60 349 
8.60 662 



8.60 973 

8.61 282 
8.61 589 
8.61 893 
8-62 196 



8.62 496 

8.62 795 

8. 63 091 
8. 63 385 
8-63 677 

8.63 968 

8.64 256 

8.64 543 
8. 64 827 

8.65 110 



8.65 391 
8.65 670 

8.65 947 

8.66 223 
8-66 497 _ 

8.66 769 
8-67 039 

8.67 308 

8.67 575 
8-6 7 840 

8.68 104 
8. 68 366 
8.68 627 
8. 68 886 
8. 69 144 



8. 69 400 

8. 69 654 
8. 69 907 

8. 70 159 
8. 70 409 



8. 70 657 

8.70 905 

8.71 150 
8-71 395 
8-71 638 
8 71 880 
Log, Cos. 



360 
357 
354 
351 

348 
346 
343 
340 
338 

335 
332 
330 
327 
325 
323 
320 
318 
316 
313 

311 
30? 
306 
304 
302 
300 
298 
296 
294 
292 

290 
288 
286 
284 
282 
281 
279 
277 
275 
274 
272 
270 
268 
267 
265 
264 
262 
260 
259 
257 
256 
254 
253 
25l 
250 
248 
247 
245 
244 
243 
241 



Log. Tan. 



57 787 

58 121 
58 451 

58 779 

59 105 



54 308 

54 669 

55 027 
55 381 
55 733 



56 083 
56 429 

56 772 

57 113 
57 452 



59 428 

59 74? 

60 067 
60 384 
60 698 



61 009 
61 319 
61 626 

61 931 

62 234 



62 535 

62 834 

63 131 
63 425 
63 718 



64 009 
64 298 
64 585 

64 870 

65 153 



65 435 
65 715 

65 993 

66 26? 
66 543 



66 816 

67 087 
67 356 
67 624 
67 890 



68 154 
68 417 
68 678 

68 938 

69 196 



69 453 
69 708 

69 961 

70 214 
70 464 



Com, D, 



70 714 

70 962 

71 208 
71 453 
71 697 



8-71 939 
Log. Cot. 



360 
358 

354 
352 

349 
346 
343 
341 
338 

335 
333 
330 
328 
325 
323 
320 
318 
316 
314 

311 
309 
307 
305 
303 
300 
299 
297 
294 
293 
291 
288 
287 
285 
283 
28l 
280 
278 
276 
274 
272 
271 
26? 
267 
266 
264 
262 
261 
259 
258 
256 
255 
253 
252 
250 
249 
248 
246 
245 
243 
242 

Com. D, 



Log. Cot, 



45 691 
45 331 
44 973 
44 618 
44 266 



Log. Cos. 



9. 99 973 
9.99 973 
9-99 972 
9-99 972 
9. 99 971 



43 917 
43 571 
43 227 
42 886 
42 548 



42 212 
41 87? 
41 548 
41 220 
40 895 



40 571 
40 251 
39 932 
39 616 
39 302 



38 990 
38 681 
38 374 
38 068 
3 7 765 
37 465 
37 166 
36 86? 
36 574 
36 281 



35 990 
35 702 
35 414 
35 12? 
34 846 



34 565 
34 285 
34 007 
33 731 
33 456 



33 184 
32 913 
32 643 
32 376 
32 110 



31 845 
31 583 
31 321 
31 062 
30 803 



30 547 
3Q 292 
30 038 
29 786 
29 535 



29 286 
29 038 
28 791 
28 546 
28 303 



28 060 



Log. Tan. 



9-99 971 
9.99 971 
9.99 970 
9-99 970 
9.99 969 



9.99 96? 
9-99 968 
9.99 968 
9.99 967 
9.99 967 



9.99 966 
9.99 966 
9.99 965 
9.99 965 
9.99 964 



9.99 964 
9. 99 963 
9-99 963 
9-99 962 
9. 99 962 



9.99 961 
9.99 961 
9.99 960 
9.99 959 
9.99 959 



9.99 958 
9.99 958 
9-99 957 
9.99 957 
9.99 956 



9.99 956 
9.99 955 
9-99 954 
9-99 954 
9.99 953 



99 953 
99 952 
99 952 
99 951 
99 950 



9. 99 950 
9. 99 94? 
9. 99 948 
9. 99 948 
9. 99 947 



9-99 947 
9-99 946 
9-99 945 
9-99 945 
9.99 944 



9.99 943 
9.99 943 
9. 99 942 
9. 99 942 
9.-99 941 



9. 99 940 
Log. Sin. 



60 

59. 
58J 
57 
56 



93** 



594 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



176* 



Log. Sin, 



8 




71 880 

72 120 
72 359 
72 597 
72 833 



75 353 
75 574 

75 795 

76 015 
76 233 



76 451 
76 667 

76 883 

77 097 
77 310 



77 522 
77 733 

77 943 

78 152 
78 360 



78 567 
78 773 

78 978 

79 183 
79 386 



79 588 
79 789 
79 989 



240 
239 
237 
236 
235 
233 
233 
231 
230 
229 
227 
226 
225 
224 
223 
221 
221 
219 
218 

217 
216 
215 
214 
213 
212 
211 
210 
209 
208 
207 
206 
205 
204 
203 
202 
201 
200 
199 



8.71 939 
8-72 180 

8.72 420 
72 659 
72 896 



80 189 TQo 
80 387' ^" 



80 585 III 
80 782|5^ 
80 977|52 
81172 J^^ 
81366 ^^_ 
^193 



81 560 



192 



81 752 .QT 
81 943!t^{ 



82 134 
82 324 



82 513 
82 701 

82 888 

83 075 
83 260 



83 445 
83 629 
83 813 



189 
189 
188 
187 
186 
185 
185 
184 
183 



84_lZ|i 
84 358 ^^^ 
Log. Cos.j d. 



Log. Tan. c.d. Log. Cot. Log. Cos 



73 131! 
73 366 



73 599 

73 831 

74 0621 



241 
240 
238 
237 
235 
K,235 
:§233 



74 292 
74 520^^.. 



232 
231 
229 
228 

1227 



74 9741 



75 199 



225 



75 422 
75 645 



:^'223 



75 867 

76 087 
76 306 



223 

221 

V 220 

- 91 Q 



76 524 



218 



76 74:1 iH 

76 958 

77 172 
77 386 



77 599 

77 811i 

78 022 
78 232 
78 441 



78 648 

78 855 

79 061 
79 266 
79 470 



79 673 

79 875 

80 076 
80 276 
80 476 



80 674 

80 871 

81 068 
81 264 
81 459 



81 653 

81 846 

82 038 
82 230 
82 420 




84 464 



Log. Cot 



216 
214 
,214 

'213 
212 
'210 
210 
209 
207 
207 
206 
204 
204 
203 
202 
20l 
200 
199 
198 
197 
197 
195 
195 
194 
193 
192 
191 
190 
190 
188 
188 
187 
186 
185 
185 
184 
183 
182 
182 

c?d 



28 060 
27 819 
27 579 
27 341 
27 104 



26 868 
26 633 
26 400 
26 168 
25J*37 

25 708 
25 479 
25 252 
25 026 
24 801 
24 577 
24 354 
24 133 
23 913 
23 693 



23 475 
23 258 
23 042 
22 827 
22 613 



22 400 
22 188 
21 978 
21 768 
21 559 



21 351 
21 144 
20 938 
20 734 
20 530 



20 327 
20 125 
19 923 
19 723 
19 524 



19 326 
19 128 
18 931 
18 736 
18 541 



18 347 
18 154 
17 961 
17 770 
17 579 



17 389 
17 201 
17 012 
16 825 
16 638 



16 453 
16 268 
16 083 
15 900 
15 717 



115 535 
Log, Tan. 



99 940 
99 940 
99 939 
99 938 
99 938 



99 937 
99 936 
99 935 
99 935 
99 934 



99 933 
99 933 
99 932 
99 931 
99 931 



99 930 
99 929 
99 928 
99 928 
99 927 



99 926 
99 925 
99 925 
99 924 
99 923 



99 922 
99 922 
99 921 
99 920 
99 919 



99 919 
99 918 
99 917 
99 916 
99 916 



99 915 
99 914 
99 913 
99 912 
99 912 



99 911 
99 910 
99 909 
99 908 
99 907 



99 907 
99 906 
99 905 
99 904 
99 903 



99 902 
99 902 
99 901 
99 900 
99 899 



99 898 
99 897 
99 896 
99 896 
99 895 



30 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

_6_ 

5 

4 

3 

2 

1 



P. P. 



9 99 894 
Log. Sin 





330 


320 


310 


6 


33.0 


32.0 


31.0 


7 


38.5 


37.3 


36.1 


8 


44.0 


42.6 


41.3 


9 


49.5 


48-0 


46-5 


10 


55.0 


53.3 


51.6 


20 


110.0 


106.6 


103.3 


30 


165.0 


160.0 


155.0 


40 


220.0 


213.3 


206.6 


50 


275.0 


266.6 


258.3 



300 

30.0 

35.0 

40.0 

45.0 

50.0 

100. 

150.0 

200.0 

250.0 





290 


280 


270 


260 


6 


29.0 


28.0 


27.0 


26.0 


7 


33.8 


32.6 


31.5 


30 


3 


8 


38.6 


37.3 


36. 


34 


(^ 


9 


43.5 


42.0 


40.5 


39 





10 


48.3 


46.6 


45.0 


43 


3 


20 


96.6 


93.3 


90.0 


86 


6 


30 


145.0 


140.0 


135.0 


130 





40 


193.3 


186.6 


180.0 


173 


3 


50 


241.6 


233.3 


225.0 


216 


6 





250 


240 


230 


22( 


6 


25.0 


24.0 


23.0 


22 


7 


29.1 


28.0 


26.8 


25. 


8 


33.3 


32.0 


30.6 


29. 


9 


37.5 


36.0 


34.5 


33. 


10 


41.6 


40.0 


38.3 


36. 


20 


83.3 


80.0 


76.6 


73- 


30 


125.0 


120.0 


115.0 


110. 


40 


166.6 


160.0 


153.3 


146. 


50 


208.3 


200.0 


191.6 


183. 





210 


200 


190 


6 


21.0 


20.0 


19.0 


7 


24.5 


23.3 


22.1 


8 


28.0 


26.6 


25.3 


9 


31.5 


30.0 


28.5 


10 


35.0 


33.3 


31.6 


20 


70.0 


66.6 


63.3 


30 


105.0 


100.0 


95.0 


40 


140.0 


133.3 


126.6 


50 


175.0 


166.6 


158.3 



180 

18. 
21.0 
24.0 
27.0 
30.0 
60.0 
90.0 
120.0 
150.0 



9 

0.9 
1.1 
1-2 
1.4 
1.6 
3.1 
4.7 
6.3 
7.9 

5 

0.4 
0.5 
0.6 
0.7 
0.7 
1.5 
2.2 
3.0 
3.7 



9 

0.9 
1.0 
1.2 
1.3 
1.5 
3.0 
4.5 
6.0 
7.5 

4 

0-4 
0.4 
0.5 
0.6 
_ 
1-3 
2.0 
2-6 
3-3 



7 
0.7 
0.8 
0.9 
1.0 
l.I 
2.3 
3.5 
4 
5.8 

2 

0.2 
0-2 
0.2 
0-3 
0.3 
0-6 
10 
1.3 
1 



6 

0.6 
0.7 
8 
09 
1.0 
2.0 
3.0 
4.0 



5 

05 
0. 



6 
7 
8 
6 
5 
3.3 



595 



5.0I4.I 
1 o 

1.0 
1.0 
1.0 
M 
1-1 

'I 

1.2 
1.3 
1.4 



86*' 






1 


0. 







0. 







0. 







0. 







0. 





3 








5 








6 








8 






TABLE VI I. 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



175' 



Log. Sin. d. Log. Tan. c.d. Log. Co' 



84 358 
84 538 
84 718 

84 897 

85 075 



85 252 
85 429 
85 605 

85 780 
85_954 

86 128 
86 301 
86 474 
86 645 
86 81 



86 987 

87 156 
87 325 
87 494 
87 661 



87 828 

87 995 

88 160 
88 326 
88 490 



88 654 
88 817 

88 980 

89 142 
89 303 



89 464 
89 624 
89 784 

89 943 

90 lOJ 

90^259 
90 417 
90 573 
90 729 
90 885 




92 561 
92 710 

92 858 

93 007 
93 154 



93 301 
93 448 
93 594 
93 740 
93 885 



8-94 029 



Log. Cos 



180 

180 

178 

178 

177 

176 

176 

175 

174 

174 

173 

172 

171 

171 

170 

169 

169 

1 

167 

167 

166 

165 

165 

164 

163 
163 
162 
162 
161 
161 
160 
159 
159 
158 
158 
157 
156 
156 
156 
155 
154 
154 
153 
153 
152 
151 
151 
150 
150 
150 
149 
148 
148 
147 

147 
146 
146 
146 
145 
144 



8.84 464 
84 645 




84 826 

85 005 
85 184 



87 106 
87 277 
87 447 
87 616 
87 785 



87 953 

88 120 
88 287 
88 453 
88 618 




91 184 
91 340 
91 495 
91 649 
91 803 



91 957 

92 109 
92 262 
92 413 
92 565 



92 715 

92 866 

93 015 
93 164 
93 313 



93 461 
93 609 
93 756 

93 903 

94 049 



894 195 



Log. Cot. 



181 

180 

179 

179 

178 

177 

176 

176 

175 

175 

174 

173 

172 

172 

171 

170 

170 

169 

1 

168 

167 

167 

166 

165 

165 

164 

163 

163 

16^ 

162 

161 

161 

160 

159 

158 
158 
157 
157 
156 

156 
155 
155 
154 
154 

153 
152 
152 
151 
151 
150 
150 
149 
149 
149 
148 
148 
147 
146 
146 
145 



c.d. 



94° 



1.15 535 
1.15354 
1.15 174 
1.14 994 
1^14 815 
1.14 637 
1.14 459 
1.14 283 
1.14 107 
1.13 931 



1.13 756 



13 582 

13 409 

1.13 237 

1.13 065 



1.12 893 
1.12 723 
1.12 553 
1.12 384 
1-12 215 



1.12 047 
1.11 880 
1.11 713 
1.11 547 
1.11 381 



11 216 
11 052 
10 889 
10 726 
10 563 



10 401 
10 240 
10 079 
09 915 
09 760 



1.09 601 
1-09 443 
1.09 285 
1.09 128 
] 08 971 



1.08 815 
1-08 660 
108 505 
1.08 350 
1.08 196 



1.08 043 
1-07 890 
1.07 738 
1.07 586 
1. 07 435 



1-07 284 
1.07 134 
1.06 984 
1.06 835 
1.06 686 



1.06 538 
1.06 390 
1-06 243 
1.06 097 
1,05 950 
1-05 805 
Log. Tan. 



Log. Cos, 



99 894 
99 893 
99 892 
99 891 
99 890 



99 889 
99 888 
99 888 
99 887 
99 886 



99 885 
99 884 
99 883 
99 882 
99 881 



99 880 
99 879 
99 878 
99 877 
99 876 



99 875 
99 874 
99 874 
99 873 
99 872 



99 871 
99 870 
99 869 
99 868 
99 867 
99 866 
99 865 
99 864 
99 863 
99 862 



99 861 
99 860 
99 R59 
99 858 
99 857 



99 856 
99 855 
99 853 
99 852 
99 851 



99 850 
99 849 
99 848 
99 847 
99 846 



99 845 
99 844 
99 843 
99 842 
99 841 



99 840 
99 839 
99 837 
99 836 
99 835 



9. 99 834 
Log. Sin. 

596 



P. P. 





181 


180 


178 


17 


ft 


6 


18.1 


18.0 


17.8 


17.6 


7 


21 


. 


21.0 


20.7 


20 


5 


8 


24 


1 


24.0 


23.7 


23 


4 


9 


27 




27.0 


26.7 


26 


4 


10 


30 




30.0 


29-6 


29 


3 


20 


60 


3 


60.0 


59.3 


58 


6 


30 


90 


5 


90-0 


89.0 


88 





40 


120 


6 


120.0 


118.6 


117 




50 


150 


8 


150.0 


148.3 


146 


6 



174 

17-4 

20 

23 

26 

29 

58 

87 
116 
145 



172 

17.2 
20 



22 
25 
28 
57 
86 
114 
143 



170 

17-01 
19 



22 
25 
28 
56 
85 
113 
141 



168 

16. 8i 
19 
22 
25 
28 
56 
84 
112 



6 
5 
3 
6 

3 
61140 





166 


164 


162 


160 


6 


16-6 


16.4 


162 


16.0 


7 


19 


3 


19 


1 


18 


9 


18 


6 


8 


22 


1 


21 


8 


21 


6 


21 


3 


9 


24 


9 


24 


6 


24 


3 


24 





10 


27 


6 


27 


3 


27 





26 


6 


20 


55 


3 


54 


6 


54 





53 


3 


30 


83 





82 





81 





80 





40 


110 




109 


3 


108 





106 


6 


50 


138 


3 


136 


6 


135 





133 


3 





158 


156 


154 


152 


6 


15-8 


15.6 


15.4 


15-2 


7 


18 


4 


18 


2 


17 


9 


17 


7 


8 


21 





20 


8 


20 


5 


20 


2 


9 


23 


7 


23 


4 


23 


1 


2"^ 


8 


10 


26 


3 


26 





25 


6 


25 


3 


20 


52 


6 


52 





51 


3 


50 


6 


30 


79 





78 





77 





76 





40 


105 


3 


104 


C 


102 


6 


101 


3 


50 


131 


6 


130 





128 


3 


126 


6 





150 


149 


148 


14 


6 


15. 


14.9 


14.8 


14 


7 


17 


5 


17 


4 


17 


2 


17 


8 


20 





19 


8 


19 


7 


19 


9 


22 


5 


22 


3 


22 


2 


22 


10 


25 





24 


3 


24 


6 


24 


20 


50 





49 


6 


49 


3 


49 


30 


75 





74 


5 


74 





73 


40 


100 





99 


3 


98 


6 


98 


50 


125 





124 


1 


123 


3 


122. 



146 

14 
17 
19 
21 
24 
48 
73 
97 
121 



145 

5 
9 
3 
7 
1 
3 
5 
6 



6 


14. 





16- 


4 


19- 


9 


21. 


3 


24- 


-6 


48. 


-0 


72 


.3 


96 


-6 


120 



1 1 

O-IjO. 
0.2 0. 
0.2I0. 
0-2|0. 
0-2i0. 
0.5|0. 
0.70. 
l.QO. 
1.210- 



O , 

1'0-Oi 

I'O-O 

10-0 

I'O.l 

10-1 

3 0.1 

5 0-2 

6 3 
80-4 



P. P. 



HI 



TABLE VII - 



-LOGARITHMIC SINES. COSINES, TANGENTS, 
AND COTANGENTS. 



17 1' 




P.P. 





145 


144 


143 14.2 


6 


14.5 


14.4 


14.3 14.2 


7 


16-9 


16 


8 


16.7 16.5 


8 


19-3 


19 


2 


19.0 18.9 


9 


21.7 


21 


6 


21.4 21-3 


10 


24.1 


24 





23.8 23-6 


20 


48.3 


48 





47.6 47.3 


30 


72.5 


72 





71.5 71.0 


40 


96.6 


96 





95.3 94.6 


50 


120.8 


120 





119.1 118-3 



141 

14.1 
16.4 
18-8 
21-1 
23.5 
47-0 
70.5 
94.0 





140 


139 


138 


137 


136 


6 


14-0 


13-9 


13.8 


13.7 


13.6 


7 


16.3 


16 


2 


16 


1 


16.0 


15 


8 


8 


18-6 


18 


5 


18 


4 


18.2 


18 


1 


9 


21.0 


20 


8 


20 


7 


20.5 


20 


4 


10 


23-3 


23 


1 


23 





22.8 


22 


6 


20 


46.6 


46 


3 


46 





45.6 


45 


3 


30 


70.0 


69 


5 


69 





68.5 


68 





40 


93.3 


92 


6 


92 





91-3 


90 


6 


50 


116.6 


115 


8 


115 





114.1 


113 


3 





135 


134 


133 


6 


13.5 


13.4 


13-3 


7 


15.7 


15.6 


15.5 


8 


18-0 


17.8 


17-7 


9 


20-2 


20.1 


19.9 


10 


22-5 


22.3 


22.1 


20 


45.0 


44-6 


44-3 


30 


67.5 


67-0 


66-5 


40 


90.0 


89-3 


88-6 


50 


112.5 


111-6 


110-8 





131 


130 


139 


6 


13-1 


13-0 


12.9 


7 


15-3 


15-1 


15.0 


8 


17-4 


17-3 


17.2 


9 


19-6 


19-5 


19.3 


10 


21.8 


21-6 


21.5 


20 


43.6 


43-3 


43-0 


30 


65-5 


65-0 


64-5 


40 


87.3 


86-6 


86-0 


50 


109.1 


108-3 


107.5 



133 

13.2 
15.4 
17.6 
19.8 
22.0 
44.0 
660 
88.0 
IIO.Q 

128 

12.8 
14.9 
17.0 
19-2 
21.3 
42.6 
64.0 
85-3 
106.6 





127 


126 


125 


124 


123 


6 


12-7 


12-6 


12-5 


12.4 


12-3 


7 


14-8 


14 


7 


14-6 


14.4 


14 


3 


8 


16-9 


16 


8 


16-6 


16.5 


16 


4 


9 


19-0 


18 


9 


18-7 


18.6 


18 


4 


10 


21.1 


21 





20-8 


20.6 


20 


5 


20 


42-3 


42 





41-6 


41.3 


41 





30 


635 


63 





62-5 


62.0 


61 


5 


40 


84-6 


84 





83-3 


82.6 


82 





50 


105-8 


105 





104-1 


103.3 


102 


5 



122 

12.2 
14.2 
16.2 
18-3 
20.3 
40.6 
61.0 
81.3 
101.6 



121 

12.1 
14.1 
16.1 
18.1 
20.1 
40.3 
60.5 
80.6 
100.8 



120 

12.0 
14.0 
16-0 
18.0 
20.0 
40.0 
60.0 
80.0 
100.0 



o 

0-9 
0-5 
0.0 
0.1 
0.1 
O.I 
0.5 
0.3 
0.4 



p.p. 



96^ 



597 



84' 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGiiiiNTS, 
AND COTANGENTS. 



173^ 



Log. Sin. 



O 

1 

2 

3 
j^ 

5 

6 

7 

8 
J_ 
10 
11 
12 
13 
lA. 
15 
16 
17 
18 
19 



9-01 
02 



1.03 
1.03 
1.03 
1.04 
1.04 



20 

21 
22 
23 
21 
25 
26 
27 
28 
29 



30 

31 

32 

33 

S4_ 

35 

36 

87 

38 

39. 

40 

41 

42 

43 

41 

45 

46 

47 

48 

4?_ 

60 

51 

52 

53 

51 

55 

56 

57 

58 

59_ 

60 



923 
043 
163 
282 
401 
520 
638 
756 
874 
992 
109 
225 
342 
458 
574 

689 
805 
919 
034 
148 
262 
376 
489 
602 
715 
828 
940 
052 
163 
275 
386 
496 
607 
717 
827 
936 
046 
155 
264 
372 
480 
588 
696 
803 
910 
017 
124 
230 
336 
442 
548 
653 
758 
863 
967 
072 
176 
279 
383 
486 
589 



Log. Cos, 



Log. Tan, 



12€ 
119 
119 
119 
119 
118 
118 
118 
117 
117 
116 
116 
116 
116 
115 
115 
114 
114 
114 
114 
113 
113 
113 
113 
112 
112 
112 
111 
111 

111 
110 
110 
110 
110 
109 
109 
109 
109 
108 
108 
108 
107 
107 
107 
107 
106 
106 
106 
106 
105 
105 
105 
104 
104 
104 
104 
103 
103 
103 
103 



9.02 162 
9.02 283 
9-02 404 
9.02 525 
9.02 645 



C.d. 



Log. Cot, 



9.02 765 

9.02 885 

9.03 004 
9.03 123 
9-03 242 



9.03 361 
9-03 479 
9. 03 597 
9.03 714 
9.03 831 



9.03 948 

9.04 065 
9.04 181 
9.04 297 
9.04 413 



9-04 528 
9-04 643 
9.04 758 
9.04 872 
9.04 987 



9.05 101 
9.05 214 
9.05 32^ 
9.05 440 
9.05 553 



9.05 666 
9.05 778 

9.05 890 

9.06 OOl 
.9-06 113 



9.06 224 
9.06 335 
9.06 445 
9.06 555 
9-06 665 



9.06 775 
9.06 884 

9.06 994 

9.07 102 
9-07 211 



9.07 319 
9.07 428 
9.07 535 
9.07 643 
9.07 750 



9.07 857 

9.07 965 

9.08 071 
9.08 177 
9.08 283 



9.08 389 
9.08 494 
9.08 600 
9.08 705 
9.08 810 



9-08 914 



121 
121 
120 
120 
120 
119 
119 
119 
119 
118 
118 
118 
117 
117 

117 
116 
116 
116 
115 
115 
115 
114 
114 
114 
114 
113 
113 
113 
113 
.112 
112 
112 
111 
111 
111 
111 
110 
110 
110 
109 
109 
109 
109 
109 
108 
108 
107 
107 
107 
107 
107 
106 
106 
106 
105 
105 
105 
105 
105 
104 



0.97 838 
0.97 716 
0.97 595 
0.97 475 
0.97 354 



Log. Cos 
9.99 76lt60 



0.97 234 
0.97 115 
0.96 995 
0.96 876 



9.99 760 
9. 99 759 
9.99 757 
9-99 756 



9.99 754 
9-99 753 
9-99 752 
9-99 750 



0.96 757I9-99 749 



0-96 639 9-99 748 
0.96 521 9-99 746 



0.96 403 
0.96 285 
0.96 168 



9-99 745 
744 
9. 99 742 



0.96 051 
0.95 935 
0.95 818 
0.95 702 
0-95 587 



0-95 471 
0.95 356 
0.95 242 
0.95 127 
0-95 013 



0.94 899 
0.94 785 
0.94 672 
0.94 559 
0-94 446 



0.94 334 
0.94 222 
0.94 110 
0.93 998 
0.93 887 



0.93 776 
0.93 665 
0-93 554 
0-93 444 
0-93 3o4 



0.93 22o 
0.93 115 
0.93 006 
0.92 897 
0.92 788 



0-92 680 
0-92 572 
0.92 464 
0.92 357 
0-92 249 



Log. Cot. 



c.d. 



0.92 142 
0.92 035 
0.91 929 
0.91 822 
0.91 716 



19 741 
9-99 739 
9.99 738 
9-99 737 
9-99 735 




9-99 720 
9.99 718 
9.99 717 
9.99 715 
9. 99 714 



9.99 712 
9. 99 711 
9-99 710 
9-99 708 
9-99 707 



9.99 705 
9.99 704 
9.99 702 
9.99 70i 
9.99 699 



9.99 698 
9.99 696 
9.99 695 
9-99 693 
9-99 692 



0.91 611 
0.91 505 
0.91 400 
0.91 295 
0.91 190 



0.91 085 
Log. Tan, 



9.99 690 
9.99 689 
9.99 687 
9-99 686 
9.99 684 



9.99 883 
9.99 681 
9-99 679 
9-99 678 
9-99 676 
9-99 675 



P.P. 



96'' 



Log. Sin, 
598 



121 


121 


120 


119 


118 


12.1 


12.1 


12.0 


11.9 


11.8 


14.2 


14.1 


14.0 


13.9 


13.7 


16.2 


16.1 


16.0 


15.8 


15.? 


18.2 


18.1 


18.0 


17-8 


17.7 


20.2 


20-1 


20.0 


19.8 


19.6 


40.5 


40-3 


40.0 


39-6 


39.^ 


60.7 


60.5 


60.0 


59.5 


59.0 


81.0 


80-6 


80.0 


79.3 


78.6 


101.2 


100.8 


100.0 


99-1 


98.3 



117. 

6:11.7 

7 13.7 

8 15-6 

9 17-6 
10 19-6 
2039.1 
30 58.7 
40 78. 3 
50 97.9 



117 116 115 

11-7 11.611.5 
13.6 13-5 13-4 
15.6 15.415.3 
17.5 17.417.2 
19-5 19.3'19.I 
39. 38.6 38-3 
58.5 58.0 57.5 
78-0 77.3 76.6 
97.5 96-6 95.8 



6 
7 
8 
9 

10 
20 
30 
40 
50 



11? 114 113 112 111 

11-2,11.1 

12.9 

9,14.8 
8 16.6 
6118.5 



11.4 
13.31 

15.2; 
17-2! 
19-1! 
38-1 
57-2 
76-3 
95-4 



11-4 
133 
15.2 
17-1 
19-0 
38-0 
57-0 
76-0 



11-3 
13-2 
15-0 
16-9 
18-8 
37-6 
56-5 
75-3 



9501941 



S37.O 

55.5 

6 74.0 
3I92.5 



110 110 

11-0 11-0 
12-9 12-8 
147 14-6 
16-6 16-5 
18-4 18-3 
36-8 36-6 
5o-2 5D-0 
73-6,73-3 
92-ll9x-6 



109 108 

10.9 10. 8 
12.7 12.6 
14-5 14.4 
16-3 16 2 
18-1 18-0 
3b. 3 360 
5^.5 5'*.0 
7-A-e 72-0 
9u-8 9u.O 



6 
7 
8 

9 
10 
20 
30 
40 
60 



107 

10.7 
12.5 
14-3 
16-1 
17-9 
35-8 
53.7 
71-6 
89.6 

lo5 



107 106 105 104 

10.4 
12.1 
13.8 
15.6 
17-3 
34-6 
52-0 
69-3 
86.6 



10.7 


10.6 


10.5 


12.5 


12.3 


12.2 


14-2 


14.1 


14.0 


16-0 


15.9 


15.7 


17-8 


17.6 


17.5 


35.6 


35-3 


35.0 


53.5 


53.0 


52.5 


71.3 


70-6 


70-0 


89.1 


88-3 


87-5 



6 
7 
8 

9 
10 
20 
30 
40 
50 



10-3 
12.1 
13-8 
15.5 
17.2 
34.5 
51.7 
690 
86.2 



103 

10.3 
12-0 
13-7 
15-4 
17-1 
34-3 
51.5 
68 _ 
85-8 



1 

0.1 
0.1 
0.1 
01 
0.1 

0.3 
0.5 

0.6 
0.8 



P.P. 



83* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



178° 



Log. Sin. I d. Log. Tan. c.d. Log. Cot. Log. Cos. 



9. 08 589 
9-08 692 
9. 08 794 
908 897 
908 999 



9. 09 101 
9 09 202 
9 09 303 
9 09 404 
9 09 505 



9-09 606 
9-09 706 
9-09 806 
9-09 906 
9-10 006 
9-10 105- 
9.10 205 
10 303 
10 402 
9-10 50li 



1029 

102 ^ 

102 

102 

102 

101 

101 

101 

101 

100 
100 
100 
100 
99 

99 
99 
98 



•10 599 
10 697 
10 795' 
10 892 
10 990 



11087 
11 184 
11 281 
11 377 
11 473 



11570 
•11 665 
•11 761 
• 11 856 
•11 952 



12 047 
12 141 
12 236 
12 330 
12 425 



9.12 518 
9.12 612 
12 706 
9 12 799 
9-12 892 



12 985 

13 078 
13 170 
13 263 

13 355 



9-13 447 
" 13 538 
9-13 630 
•13 721 
913 813 



13 903 

13 994 

14 085 
14 175 
14 265 



9-14355 



•08 914 
09 018 
09 123 
09 226 
09 330 



•09 433 
•09 536 
■ 09 639 
09 742 
09 844 



09 947 

10 048 
10 150 
10 252 
10 353 



10 454 
10 555 
10 655 
10 756 
10 856 



• 10 956 
.11055 

• 11 155 
•11 254 
•11353 



.11452 
.11 550 
11 649 
11 747 
.11 843 



.11943 
12 040 

■ 12 137 
12 235 
12 331 



■ 12 428 
12 525 

•12 621 
12 717 
12 813 



•12 908 
•13 004 
•13 099 
13 194 

• 13 289 



•13 384 
13 478 
•13 572 
•13 666 
•13 760 



•13 854 
•13 947 
•14 041 
•14 134 
•14 227 



14 319 
14 412 
14 504 
14 596 
■ 14 688 



9 14 730 



104 
104 
103 
103 
103 
103 
103 
102 
102 
102 
101 
102 
101 
101 

101 
101 
100 
100 
100 
100 

99 

99 



• 91 085 

• 90 981 

• 90 877 
.90 773 
.90 670 



.90 566 
.90 463 
.90 360 
90 258 
90 155 
90 053 
89 951 
89 849 
89 748 
89647 



■ 99 675 
■99 673 
.99 672 
.99 670 



.99 667 

■ 99 665 

■ 99 664 
.99 662 

99 661 



89 546 
89 445 
89 344 
89 244 
89 144 



89 044 
88 944 
88 845 
88 745 
88 646 



88 548 
88 449 
88 351 
88 253 
88 155 



88 057 
87 959 
87 862 
87 765 
87 668 



87 571 
87 475 
87 379 
87 283 
87 187 



87 091 
86 996 
86 900 
86 805 
86 710 



86 616 
86 521 
86 427 
86 333 
86 239 



86 146 
86 052 
85 959 
85 866 
85 773 



85 680 
85 588 
85 495 
85 403 
85 311 



0-85 219 



99 659 
99 658 
99 656 
99 654 
99 653 



.99 651 
.99 650 
.99 648 
■ 99 646 
99 645 



99 643 
99 641 
99 640 
99 638 
99 637 



99 635 
99 63 
99 632 
99 630 
99 628 



.99 627 
.99 625 
.99 623 
.99 622 
.99 620 



• 99 618 

• 99 617 
.99 615 
.99 613 

• 99 611 



.99 610 
.99 608 
.99 606 
.99 605 
-99 603 



99 601 
99 600 
99 598 
99 596 
99 594 



.99 593 
.99 591 
.99 589 
.99 587 
• 99 586 



■99 584 
99 582 
■99 580 
.99 579 
•99 577 



9-99 575 



Log. Cos. d. Log. Cot. c.d. Log, Tan. Log. Sin. 
. -..- ^. 599 



P. P. 





104 


103 


103 


6 


10-4 


10^3 


10.2 


7 


12 


1 


12 





11 


9 


8 


13 


8 


13 


7 


13 


6 


9 


15 


6 


15 


4 


15 


3 


10 


17 


3 


17 


1 


17 





20 


34 


6 


34 


3 


34 





30 


52 





51 


5 


51 





40 


69 


3 


68 


6 


68 





50 


86 


6 


85 


8 


85 








100 


100 


99 


98 


6 


10^0 


10.0 


9.9 


9. 


7 


11.7 


11 


6 


11.5 


11- 


8 


13^4 


13 


3 


13.2 


13. 


9 


15.1 


15 





14.8 


14. 


10 


16. 7 


16 


6 


16. 5 


16. 


20 


33.5 


33 


3 


33.0 


32. 


30 


50.2 


50 





49.5 


49 • 


40 


67^0 


66 


6 


66.0 


65- 


50 


83.7 


83 


8 


82-5 


81. 



97 


97 




96 , 


9.7 


9.7 


9.6I 


11 


4 


11 


3 


11 


2 


13 





12 


9 


12 


8 


14 


5 


14 


5 


14 


4 


16 


2 


16 


1 


16 





32 


5 


32 


3 


32 





48 


7 


48 


5 


48 





65 





64 


6 


64 





81 


2 


80 


8 


80 








9? 


94 


93 


6 


9.4 


9.4 


9.3 


7 


11 





10.9 


10.8 


8 


12 


6 


12.5 


12.4 


9 


14 


2 


14.1 


13.9 


10 


15 


7 


15^6 


15.5 


20 


31 


5 


31 3 


31.0 


30 


47 


2 


47^0 


46.5 


40 


63 





62^6 


620 


50 


78 


7 


78^3 


77.5 



101 

ICl 
11.8 
13.4 
15.1 
16-8 
33.6 
50.5 
67.3 
84. i 



95 

9.5 
11.1 
12-6 
14.2 
15.8 
31.6 
47.5 
63.3 
79.1 



92 

9-2 
10.? 
12.2 
13.8 
15.3 
30.6 
46-0 
61. 5 
76.6 





9T 


91 


90 


2 




6 


9.1 


9.1 


90 


0.2 


0. 


7 


10 


7 


10 


6 


10^5 


0.2 


0. 


8 


12 


2 


12 


1 


12^0 


0^2 


0. 


9 


18 




13 


6 


13-5 


0.3 


0. 


10 


15 


2 


15 


1 


15-0 


0.3 


0. 


20 


30 


5 


30 


3 


30^0 


0.6 





30 


45 


7 


45 


5 


45.0 


1.0 





40 


61 





60 


6 


60^0 


1.3 


1 


50 


76 


2 


75 


8 


75.0 


1.6 


1 



P. p. 



83' 



TABLE VII.- 



SP 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



171^ 



' Log. Sin. d. Log. Tan. c.d. Log. Cot. Log. Cos 



55 
56 
57 
58 
51 
60 



14 355 
14 445 
14 535 
14 624 
14 713 



14 802 
14 891 

14 980 

15 068 
15 157 



15 245 
15 333 
15 421 
15 508 
15 595 



15 683 
15 770 
15 857 

15 943 

16 030 



16 116 
16 202 
16 288 
16 374 
16 460 



16 545 
16 630 
16 716 
16 801 
16 885 

16 970 

17 054 
17 139 
17 223 
17 307 



17 391 
17 474 
17 558 
17 641 
17724 



17 807 
17 890 

17 972 

18 055 
18 137 



18 219 
18 301 
18 383 
18 465 
18 546 



18 628 
18 709 
18 790 
18 871 
18 952 



19 032 
19 113 
19 193 
19 273 
19 353 



19 433 



Log. Cos, 



14 780 
14 872 

14 963 

15 054 
15 145 



15 236 
15 327 
15 417 
15 507 
15 598 



15 687 
15 777 
15 867 

15 956 

16 045 
16 134 
16 223 
16 312 
16 401 
16 489 



16 577 
16 665 
16 753 
16 841 
16 928 



17 015 
17 103 
17 190 
17 276 
17 363 



17 450 
17 536 
17 622 
17 708 
17 794 



17 880 

17 965 

18 051 
18 136 
18 221 



18 306 
18 390 
18 475 
18 559 
18 644 




19 560 
19 643 
19 725 
19 807 
19 889 



19 971 
Log. Cot. 



89 



• 84 763 

■ 84 673 

• 84 582 

• 84 492 

■ 84 402 



:312 
:222 
: 133 

t043 
i 954 

I 865 
t 776 
; 687 
I 599 
i 511 



. 272 
. 188 
. 104 
. 020 
L937 
I 854 
> 770 
I 687 
I 604 
I 522 

• 80 439 
80 357 

• 80 274 

• 80 192 
80 110 



0-80 028 
Log. Tan. 



• 99 575 
99 573 
•99 571 
•99 570 
•99 568 



■ 99 566 

• 99 564 

• 99 563 
•99 561 
•99 559 



•99 557 
•99 555 

• 99 553 

• 99 552 

• 99 550 



99 548 

• 99 546 

• 99 544 

• 99 542 
99 541 



99 539 
• 99 537 
•99 535 
•99 533 
-99 531 



• 99 529 
.99 528 

• 99 526 

• 99 524 
99 522 



•99 520 
■ 99 518 
•99 516 
• 99 514 
99 512 



• 99 511 

• 99 509 
99 507 
99 505 
99 503 



■ 99 501 

■ 99 499 

■ 99 497 
-99 495 

99 493 



99 482 
99 480 

■ 99 478 
99 476 

• 99 474 



■ 99 472 
99 470 

■ 99 468 
-99 466 

99 464 



9-99 462 
Log. Sin, 



P. P. 





91 


91 


90 


8J 


6 


9.1 


9-1 


90 


8. 


7 


10 


7 


10 


6 


10 


5 


10 


8 


12 


2 


12 


1 


12 





11. 


9 


13 


7 


13 


6 


13 


5 


13- 


10 


15 


2 


15 


1 


15 





14. 


20 


30 


5 


30 


3 


30 





29. 


30 


45 


7 


45 


5 


45 





44. 


40 


61 





60 


6 


60 





59. 


50 


76 


2 


75 


8 


75 





74. 





88 


88 


87 


8f 


6 


8 8 


8-8 


8-7 


8. 


7 


10 


3 


10 


2 


10 


1 


10. 


8 


11 


8 


11 


7 


11 


6 


11. 


9 


13 


3 


13 


2 


13 





12. 


10 


14 


7 


14 


6 


14 


5 


14. 


20 


29 


5 


29 


3 


29 





28. 


30 


44 


2 


44 





43 


5 


43. 


40 


59 





58 


6 


58 





57. 


50 


73 


7 


73 


3 


72 


5 


71 





85 


•85 


84 


83 


6 


8.5 


8.5 


8 4 


8. 


7 


10 





9 


9 


9 


8 


9. 


•8 


11 


4 


11 


3 


11 


2 


11. 


9 


12 


8 


12 


7 


12 


6 


12^ 


10 


14 


2 


14 


1 


14 





13^ 


20 


28 


5 


28 




28 





27- 


30 


42 


7 


42 


5 


42 





41. 


40 


57 





56 


6 


56 





55- 


50 


71 


2 


70 


8 


70 





69. 





82 


82 


81 


8( 


6 


8-2 


8.2 


8.1 


8. 


7 


9 


6 


9 


5 


9 


4 


9- 


8 


11 





10 


9 


10 


8 


10- 


9 


12 


4 


12 


3 


12 


1 


12 


10 


13 


7 


13 


6 


13 


5 


13 


20 


27 


5 


27 


3 


27 





26- 


30 


41 


2 


41 





40 


5 


40 


40 


55 





54 


6 


54 





53. 


50 


68 


7 


68 


3 


67 


5 


66. 



10 
20 
30 
40 
50 



7 


9 


2 


1 


7-9 


0-2 





9 


3 


2 





10 


6 


0-2 





11 


9 


0-3 





13 


2 


0^3 





26 


5 


0.6 





39 


7 


1.0 


0. 


53 





1.3 


1. 


66 


2 


1.6 


1 



p.p. 



98^ 



600 



81** 



TABLE VII.— i^OGARITHMIC SINES, COSINES. TANGENTS, 

AND COTANGENTS. 170** 



' Log. Sin. 



19 433 
19 513 
19 592 
19 672 
19 75] 



19 830 
19 909 

19 988 

20 066 
20145 



20 223 
20 301 
20 379 
20 457 
20^35 
20 613 
20 690 
20 768 
20 845 
20 922 



20 999 

21 076 
21 152 
21 229 
21 305 



21 382 
21 458 
21 534 
21 609 
21 685 



21 761 
21 836 
21 911 

21 987 

22 062 



22 136 
22 211 
22 286 
22 360 
22 435 



22 509 
22 583 
22 657 
22 731 
22 805 



22 878 

22 952 

23 025 
23 098 
23 171 



23 244 
23 317 
23 390 
23 462 
23 535 



23 607 
23 679 
23 751 
23 823 
23 895 



23 967 



Log. Cos 



9 971 

9-20 053 

20 134 

20 216 

20 297 



Log. Tan. c.d 



20 378 
20 459 
20 540 
20 620 
20 701 



20 781 
20 862 

20 942 

21 022 
21 102 



21 181 
21 261 
21 340 
21 420 
21 499 



21 578 
21 657 
21 735 
21 814 
21 892 



21 971 

22 049 
22 127 
22 205 
22J83 

22 365 
22 438 
22 515 
22 593 
22 670 



22 747 
22 824 
22 900 

22 977 

23 054 



23 130 
23 206 
23 282 
23 358 
23 434 



23 510 
23 586 
23 66l 
23 737 
23 812 



23 887 

23 962 

24 037 

24 112 
24 186 



24 261 
24 335 
. 24 409 
9-24 484 
9-24 558 
9-24 632 



81 
81 
81 
81 
81 
! 81 
81 
80 
81 
80 
80 
80 
80 
80 

79 
79 
79 
79 
79 
79 
79 
78 
78 
78 
78 
78 
78 
78 
78 
77 
77 
77 
77 
77 

77 
77 
76 
77 
76 
76 
76 
76 
76 
76 
76 
75 
75 
75 
75 

75 
75 
75 
75 
74 
74 
74 
74 
74 
74 
74 



Log. Cot. Log. Cos. 



80 02 
79 947 
79 865 
79 784 
79 703 



79 622 
79 541 
79 460 
79 379 
79 298 



79 218 
79 138 
79 058 
78 978 
78 898 



78 818 
78 739 
78 659 
78 580 
78 501 



78 422 
78 343 
78 264 
78 186 
78 107 




870 
793 
717 
641 
565 



76 489 
76 414 
76 338 
76 263 
76 188 



76 113 
76 038 
75 963 
75 888 
75 813 



75 739 
75 664 
75 590 
75 516 
75 442 



75 368 



99 462 
99 460 
99 458 
99 456 
99 454 
99 452 
99 450 
99 448 
99 446 
99 444 



99 442 
99 440 
99 437 
99 435 
99 433 



99 431 
99 429 
99 427 
99 425 
99 428 



99 421 
99 419 
99 417 
99 415 
99 413 



99 411 
99 408 
99 406 
99 404 
99 402 



99 400 
99 398 
99 396 
99 394 
99 392 



99 389 
99 387 
99 385 
99 383 
99 381 



99 379 
99 377 
99 374 
99 372 
99 370 



40 

39 
38 
37 
36_ 
35 
34 
33 
32 
IL 
30 
29 
28 
27 
26. 
25 
24 
23 
22 
21 



99 368 
99 366 
99 364 
99 361 
99 359 



99 357 
99 355 
99 353 
99 350 
99 348 



99 346 
99 344 
99 342 
99 339 
99 337 



9-99 335 



Log. Cot. c.d.iLog. Tan. Log. Sin 
601 



P. P. 



81 81 80 



1 
5 

10 
2il2 
6|13 
1127 
7;40 
3 54 
9167 



79 

7-9 



78_ 78 



9 

10 : 
20 : 
30 : 

40 ! 
50 ( 



7 


8 


7 


8 


7- 


9 


]. 


9 


1 


9. 


10 


4 


10 


4 


10. 


11 


8 


11 


7 


11. 


13 


1 


13 





12. 


26 


1 


26 





25. 


39 


o 


39 





38. 


52 


3 


52 





51. 


65 


4 


65 





64 



77 
7 





7^ 




76 


75 


74 


6 


7 6 


7.6 


7-5 


7. 


7 


8 


9 


88 


8 


7 


8. 


8 


10 


2 


101 


10 





9. 


9 


11 


5 


11-4 


11 


2 


11. 


10 


12 


7 


12-6 


12 


5 


12. 


20 


25 


5 


25.3 


25 





24. 


30 


38 


2 


380 


37 


5 


37- 


40 


51 





50.6 


50 





49 


50 


63 


7 


63.3 


62 


5 


61 





73 


73 


72 




6 


7 3 


73 


7.2 


7 


8 


6 


85 


8 


4 


8 


9 


8 


97 


9 


6 


9 


11 





10. 9 


10 


8 


10 


12 


2 


12-1 


12 





20 


24 


5 


24-3 


24 





30 


36 


7 


36.5 


36 





40 


49 





48 6 


48 





50 


61 


2 


60.8 


60 








71 




71 


2 , 


6 


7.1 


7.1 


0-2| 


7 


8 


3 


8 


3 





3 


8 


9 


5 


9 


4 





3 


9 


10 


7 


10 


6 





4 


10 


11 


9 


11 


8 





4 


20 


23 


8 


23 


6 





8 


30 


35 


7 


35 


5 


1 


2 


40 


47 


6 


47 


3 


1 


6 


50 


59 


6 


59 


1 


2 


1 



2 

0.2 
0.2 
0.2 
0.3 
0-3 
0-6 
1.0 
1-3 
1-6 



P- P. 



SO"" 



10° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. iq^^ 



Log. Sin. d. Log. Tan. c. d, Log. Cot. Log. Cos, 



23 967 

24 038 
24 110 
24 181 

24 252 



24 323 
24 394 
24 465 
24 536 
24 607 



24 677 
24 748 
24 818 
24 888 
24 958 



25 028 
25 098 
25 167 
25 237 
25 306 



25 376 
25 445 
25 514 
25 583 
25 652 



25 721 
25 790 
25 858 

25 927 
15J95 

26 061 
26 131 
26 199 
26 267 
26 335 



26 402 
26 470 
26 537 
26 605 
26 672 



26 739 
26 806 
26 873 

26 940 

27 007 



27 073 
27 140 
27 206 
27 272 
27 339 



27 405 
27 471 
27 536 
27 602 
27668 



27 733 
27 799 
27 864 
27 929 
27 995 



9 28 060 



Log. Cos 



100° 



24 632 
24 705 
24 779 
24 853 
24 926 



25 000 
25 073 
25 146 
25 219 
25292 



25 365 
25 437 
25 510 
25 582 
25 654 



25 727 
25 799 
25 871 
25 943 
26014 



26 086 
26 158 
26 229 
26 300 
263 7 1 



26 443 
26 514 
26 584 
26 655 
26 726 



26 796 
26 867 

26 937 

27 007 
27 078 



27 148 
27 218 
27 287 
27 357 

27 427 



27 496 
27 566 
27 635 
27 704 
27 773 



27 842 
27 911 

27 980 

28 049 
28 117 



Log. 



186 
254 
322 
390 
459 
527 
594 
662 
730 
797 
865 
Cot 



69 



68 



75 368 
75 294 
75 220 
75 147 
75 073 



75 000 
74 927 
74 854 
74 781 
74708 



74 635 
74 562 
74 490 
74 417 
74 345 



74 273 
74 201 
74 129 
74 057 
73 985 



913 
842 
771 
699 
628 
557 
486 
415 
344 
274 



73 203 
73 133 
73 062 
72 992 
72 922 



72 852 
72 782 
72 712 
72 642 
72 573 



72 503 
72 434 
72 365 
72 295 
72 226 



72 157 
72 088 
72 020 
71 951 
71 882 



71814 
71 746 
71 677 
71 609 
71 541 




Log. Tan. 



9-99 335 
99 333 
99 330 
99 328 
99 326 



99 324 
99 321 
99 319 
99 317 
99315 



99 312 
99 310 
99 308 
99 306 
99 303 



99 301 
99 299 
99 296 
99 294 
99 292 




99 255 
99 252 
99 250 
99 248 
99 245 



243 
240 
238 
236 
233 
231 
228 
226 
224 
221 



99 



219 
216 
214 
212 
209 
207 
204 
202 
199 
197 
194 



Log. Sin 
602 



P. p. 





74 


73 


6 


7-4 


7-3 


7 


8.6 


8.6 


8 


98 


9.8 


9 


11.1 


11.0 


10 


12-3 


12.2 


20 


24.6 


24.5 


30 


37 


36. 7 


40 


49-3 


49. 


50 


61.6 


61-2 



10.9 
12.1 
24.3 
36.5 
48. 6 
60.8 





7 


2 


72 


71 


71 


6 


7.2 


7.2 


7-1 


71 


7 


8 


4 


84 


8 


3 


8 


3 


8 


9 


6 


96 


9 


5 


9 


4 


9 


10 


9 


10.8 


10 


7 


10 


g 


10 


12 


1 


12.0 


11 




U 


3 


20 


24 


1 


24-0 


23 


s 


23 


6 


30 36 


2 


36 


35 


7 


35 


5 


40 48 


3 


48. 


47 


6 


47 




50 60 


4 


60. 


59 


6 


59 


1 





7 


[) 


7< 


D 


69 


or 


6 


7.0 


70 


6-9 


6 


7 


8 


2 


8 


1 


8 


1 


8 


8 


9 


4 


9 


3 


9 


2 


9 


9 


10 


6 


10 


5!10 


4 


10 


10 


11 


7 


11 


6 11 


6 


11 


20 


23 


5 


23 


3 23 


1 


23 


30 


35 


2 


35 


34 


7 


34 


40 


47 





46 


6 46 


3 


46 


50 


58 


7 


58 


3 57 


9 


57. 





68 


68 


67 


67 


6 


68 


6 8 


6.7 


6 


7 


8 





7 


9 


7 


9 


7 


8 


9 


1 


9 





9 





8 


9 


10 


3 


10 


2 


10 


1 


10 


10 


11 


4 


11 


3 


11 


2 


11 


20 


22 


8 


22 


6 


22 


5 


22 


30 


34 


2 


34 





33 


7 


33 


40 


45 


6 


45 


3 


45 





44. 


50 


57 


1 


56 


6 


56 


2 


55 



5 9 
J.O 
).2 
).3 

L.5 
JO 
15 
) 
75 



66 66 



65 

6 5 



•6 
■ 7 

8 

.9jl0 
• 8 21 
•7 32 

6(43 
.6154 



65 

65 



2 2 






2 


0. 





3 


0. 





3 


0. 





4 


0- 





4 


0. 





8 


0. 


1 


2 


1. 


1 


6 


1. 


2 


1 


1. 



p. p. 



79^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



168° 



Log. Sin. d 



9-28 383 
9- 28 448 
9-28 512 
9.28 576 
9.28 641 



9-28 705 
9. 28 769 
9. 28 832 
28 896 
9. 28 960 



29 023 
•29 087 
•29 150 
•29 213 

29 277 



•29 340 
•29 403 
•29 466 
29 528 
:_29_591 
•29 654 
•29 716 
•29 779 
•29 841 
•29 903 



•30 275 
• 30 336 
•30 398 
•30 459 
•30 520 



• 30 886 

• 30 947 

• 31 008 

• 31 068 
•31 129 



31 189 

9-31 249 

31 309 

31 370 

9.31 429 



9.31 489 

9.31 549 

31 609 

31 669 

9.31 728 



9-31 788 



Log. Cos. 



Tan. c. d. Log. Cot. Log. Cos. 



9. 30 846 
9.30 911 

9.30 975 
31 040 

9. 31 104 



9.31 168 
931 232 
9-31 297 
9^31 361 
31 424 



9-31 488 
9^31 552 
31 616 
9.31 679 
9.31 743 



931 806 
931 869 
9-31 933 

31 996 

32 059 



9-32 122 
9 32 ^85 
9-32 248 
9.32 310 
932 373 



32 436 
9.32 498 
9.32 560 

32 623 
9-32 685 



9 32 747 
Log. Cot. c.d. 



0-70 798 
70 732 
070 665 
0.70 598 
0.70 531 



070 465 
070 398 
0-70 332 
0.70 266 
0.70 200 



70 134 
0.70 068 
0-70 002 
0.69 936 
0-69 870 



999 182 
99 180 
999 177 
9-99 175 
9. 99 172 



9. 99 170 
9-99 167 
9.99 165 
9. 99 162 
9.99 160 



•99 157 
•99 155 
•99 152 
•99 150 
•99 147 



0-68 831 
0.68 767 
0-68 703 
068 639 
0.68575 



69 805 
069 739 
69 674 
0-69 608 
0.69 543 
69 478 9 
0.69 413 
69 348 
069 283 
0.69 218 
0.69 153 
0.69 089 
0.69 024 
0.68 960 
068 896 



•99 145 
•99 142 
•99 139 
•99 137 
•99 134 



68 511 
68 447 
68 384 
68 320 
68 257 



68 193 

68 130 

0.68 067 

0.68 004 

0.67 941 



067 878 
067 815 
0.67 752 
067 689 
0.67 626 



0-67 564 
0.67 501 
0.67 439 
0.67 377 
0.67 314 



0-67 252 
Log. Tan. 



99 132 
99 129 
99 127 
99 124 
99 122 



•99 119 
•99 116 
•99 114 
•99 111 
•99 109 



•99 106 
•99 104 
•99 101 
99 098 
•99 096 



.99 093 
•99 091 

• 99 088 

• 99 085 
•99 083 



• 99 080 
.99 077 
.99 075 

• 99 072 
-99 069 



• 99 067 

• 99 064 

• 99 062 
.99 059 
.99 056 



• 99 054 

• 99 051 
.99 048 
.99 046 
.99 043 



9-99 040 
Log. Sin. 



P. P. 



10 





6 


/ 


6' 


6 


67 


6 


7 


7 


9 


7 


8 


9 





8 


9 


10 


1 


10 


10 


11 


2 


11 


20 


22 


5 


22 


30 


33 


7 


33 


40 


45 





44 


50 


56 


2 


55 





66 


66 


65 


6.' 


6 


6.6 


6-6 


6.5 


6 


7 


7 


7 


7 


7 


7 


6 


7 


8 


8 


8 


8 


8 


8 


7 


8 


8 


10 





9 


9 


9 


8 


9 


10 


11 


1 


11 





10 


9 


10 


20 


22 


1 


22 





21 


8 


21 


30 


33 


2 


33 





32 


7 


32. 


40 


44 


3 


44 





43 


5 


43 


50 


55 


4 


55 





54 


6 


54 



6; 
7, 7 

8j 8 

9' 9 
10:10 
20121 
30|32 
4043 
50I53 



64 

64 



64 

6.4 



63_ 

3 
4 
4 
5 



63 

6 
7 





63 


62 


61 


61 


6 


6.2 


6.2 


6.1 


6 


7 


7 


3 


7 


2 


7 


2 


7 


8 


8 


3 


8 


2 


8 


2 


8. 


9 


9 


4 


9 


3 


9 


2 


9. 


10 


10 


4 


10 


3 


10 


2 


10. 


20 


20 


8 


20 


6 


20 


5 


20. 


30 


31 


2 


31 





30 


7 


30. 


40 


41 


6 


41 


3 


41 





40. 


50 


52 


1 


51 


6 


51 


2 


50. 





60 


60 


5 


5 


6 


6 


60 


5 9 


7 


7 





7 





6 


9 


8 


8 





8 





7 


9 


9 


9 


1 


9 





8 


9 


10 


10 


1 


10 





9 


9 


20 


20 


1 


20 





19 


8 


30 


30 


2 


30 





29 


7 


40 


40 


3 


40 





39 


6 


50 


50 


4 


50 





49 


6 





3 


i 


2 


6 


0.3 


0.21 


7 





3 





3 


8 





4 





3 


9 





4 





4 


10 





5 





4 


20 


1 








8 


30 


1 


5 


1 


2 


40 


2 





1 


6 


50 


2 


5 


2 


1 



0.2 
0-2 
0.2 
0.3 
0.3 
0.6 
l.Q 
1.3 
1-6 



P. P. 



lor 



603 



78** 



TABLE VII.- 



13° 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



167 • lit 



o 

1 

2 
3 

5 

6 

7 

8 
JL 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 



20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

81 

32 

33 

34 



35 

36 

87 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 



Log, Sin. d. Log. Tan, c.d, Log. Cot, Log. Cos, 



31 788 
31 847 
31 906 

31 966 

32 025 



32 084 
32 143 
32 202 
32 260 
32 319 



32 378 
32 436 
32 495 
32 553 
32^611 
32 670 
32 728 
32 786 
32 844 
32 902 




33 817 
33 874 
33 930 

33 987 

34 043 



34 099 
34 156 
34 212 
34 268 
34 324 



34 379 
34 435 
34 491 
34 547 
34 602 



34 658 
34 713 
34 768 
34 824 
34 879 



34 934 

34 989 

35 044 
35 099 
35 154 



9 35 209 



Log. Cos. 



32 747 
32 809 
32 871 
32 933 
32 995 



33 057 
33 118 
33 180 
33 242 
33 303 



33 364 
33 426 
33 487 
33 548 
33 609 
33 670 
33 731 
33 792 
33 852 
33 913 



33 974 

34 034 
34 095 
34 155 
34 215 
"34 275 

34 336 
34 396 
34 456 
34 515 



34 575 
34 635 
34 695 
34 754 
34 814 



34 873 
34 933 

34 992 

35 051 
35 110 



35 169 
35 228 
35 287 
35 346 
35 405 



35 464 
35 522 
35 581 
35 640 
35 698 



35 756 
35 815 
35 873 
35 931 
35 989 




Log. Cot, 



c.d, 



-67 252 

• 67 190 

• 67 128 
67 066 

• 67 004 

• 66 943 
66 881 

• 66 819 

■ 66 758 

• 66 696 
66 635 

■ 66 574 

• 66 513 

• 66 452 
66 390 



.99 040 

• 99 038 

• 99 035 

• 99 032 

• 9 9 029 



• 99 013 

• 99 010 

• 99 008 
99 005 

■99 002 



• 66 330 9 • 

• 66 269 9^ 

• 66 208 
■ 66 147 

66 086 



64 830 

• 64 771 

• 64 712 

• 64 653 

• 64 594 



• 64 536 

• 64 477 
.64 418 

• 64 360 

• 64 302 



■ 64 243 

■ 64 185 

• 64 127 

• 64 068 

• 64 010 



•63 952 

• 63 894 

• 63 837 
•63 779 
•63 721 



0-63 663 
Log, Tan. 



98 986 

• 98 983 

• 98 980 

• 98 977 

• 98 975 

• 98 972 
•98 969 
•98 966 

• 98 963 

• 98 961 



• 98 944 
98 941 

• 98 938 

• 98 935 

• 98 933 



• 98 930 
98 927 

• 98 924 

• 98 921 

• 98 918 



• 98 915 
98 913 
98 910 

• 98 907 

• 98 904 



• 98 901 

• 98 898 

• 98 895 

• 98 892 

• 98 890 



• 98 887 
98 884 
98 881 
98 878 

• 98 875 



103° 



98 872 

Log. Sin. 

604 



10 



P. P. 



62 

62 



61 

6 
7 
8 

9 

10 
20 
30 
41 
51 



61 

61 
71 

8 1 

9 1 
10-1 
20 3 
30-5 
40^6 
50 8 





60 


60 


59 


5fi 


6 


6^0 


60 


5.9 


5^ 


7 


7 





7 





6 


9 


6- 


8 


8 





8 





7 


9 


7- 


9 


9 


1 


9 





8 


9 


s. 


10 


10 


1 


10 





9 


9 


Q. 


20 


20 


1 


20 





19 


8 


19 


30 


30 


2 


30 





29 


7 


29 


40 


40 


3 


40 





39 


6 


39 


50 


50 


4 


50 





49 


6 


49 



58_ 
58 
6 8 
78 

8 8 

9 7| 9 
19.5 19 
29 2 29 
39^0 38 
487 48 



58 
58i 



•^7_ 

5 7 

7 

7 

8 

9 

19 



7 

7 

7 

6 

3 

0|28 

6138 

3147 



57 

5 7 





56 


56 


55 5^^ 


6 


5^6 


56 


55i 5 


7 


6 


6 


6 


5 


6 


5| 6 


8 


7 


f; 


7 


4 


7 


4 7 


9 


8 


5 


8 


4 


8 


3 8 


10 


9 


4 


9 


3 


9 


2! 9 


20 


18 


8 


18 


6 


18 


5II8 


30 


28 


2 


28 





27 


7127 


40 


37 


6 


37 


3 


37 


0I36 


50 


47 


1 


46 


6 


46 


2!45 





54 




^ 


2 


6 


54 


0-3 


0. 


7 


6 


3 





3 


0. 


8 


7 


2 





4 


0. 


9 


8 


2 





4 


0. 


10 


9 


1 





5 





2C 


18 


1 


1 





0^ 


30 


27 


2 


1 


5 


1^ 


40 


36 


3 


2 





1^ 


50 


45 


4 


2 


5 


2. 



P. K. 



77° 



T\BLE VII.—LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 




1G6° 



P. p. 



57_ 

5.7 
7 
6 



40 38 
50 47 



57 

5 

6 

7 

8 

9 
19 
28 
38 
47 



56_ 

5.61 
6! 
5 



6 37 
l'46 





55 


55 


54 


54 


6 


5.5 


5-5 


5-4 


5 


7 


65 


6 


4 


6 


3 


6 


8 


7.4 


7 


3 


7 


2 


7 


9 


8 3 


8 


2 


8 


2 


8 


10 


9.2 


9 


1 


9 


1 


9 


20 


18-5 


18 


3 


18 


1 


18 


30 


27.7 


27 


5 


27 


2 


27 


40 


37.0 


36 


6 


36 


3 


36 


50 


46. 2 


45 


8 


45 


4 


45 



56 

56 
5 
4 
4 
3 
6 

3 
6 





53 


53 


52 


52 


6 


5.3 


5.3 


5.2 


5. 


7 


6 


2 


6 


2 


6 


1 


6. 


8 


7 


1 


7 





7 





6- 


9 


8 





7 


9 


7 


9 


7. 


10 


8 


9 


8 


8 


8 


7 


8- 


20 


17 


8 


17 


6 


17 


5 


17- 


30 


26 


7 


26 


5 


26 


2 


26- 


40 


35 


6 


35 


3 


35 





34- 


50 


44 


6 


44 


1 


43 


7 


43. 



51 



7 6 



10 
20 
30 
40 
50 



50 

50 
9 
7 
6 
4 

16 8 

25-2 

33 

42 



3 

610 
70 
8|0 
9iO 
10 



20 
30 
40 
50 



3 2, 



0.8 
1.2 
1.6 
2.1 



P. P. 



103° 



605 



76^ 



14° 



TABLE VII.—LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 1^50 



Log. Sin, 




38 871 
38 921 

38 971 

39 021 
39 071 



39 120 
39 170 
39 220 
39 269 
39 319 



39 368 
39 418 
39 467 
39 516 
39 566 



39 615 
39 664 
39 713 
39 762 
39 811 



39 860 
39 909 

39 957 

40 006 
40 055 



40 103 
40 152 
40 200 
40 249 
40297 



40 345 
40 3941 
40 442! 
40 490! 
40 538 



40 586; 
40 634' 
40 682 
40 730! 
40 7771 



40 825: 
40 873: 
40 920j 

40 968; 

41 015! 



41 063 
41 110 
41 158 
41 205 
41 252 



41 299i 



Log, Tan. c, d-jLog, Cot. Log. Cos. 



Log. Cos. I d. Log. Cot.jC. d 



39 677 
39 731 
39 784 
39 838 
39 892 



39 945 

39 999 

40 052 
40 106 
40 159 



40 212 
40 265 
40 318 
40 372 
40 425 



40 478 
40 531 
40 583 
40 636 
40689 



40 742 
40 794 
40 847 
40 899 
40 952 



41 004 
41 057 
41 109 
41 16l 
41 213 



41 266 
41 318 
41 370 
41 422 
41 474 



41 525 
41 577 
41 629 
41 681 
41 732 



41 784 
41 836 
41 887 
41 938 
41 990 



42 041 
42 092 
42 144 
42 195 
42 246 



42 297 
42 348 
42 399 
42 450 
42 501 



42 552 
42 602 
42 653 
42 704 
4 2 754 
42 805 



60 323 
60 269 
60 215 
60 161 
60 108 



054 
001 
947 
894 
841 
787 
734 
681 
628 
575 



59 522 
59 469 
59 416 
59 363 
59 311 



59 258 
59 205 
59 153 
59 100 
59 048 



58 995 
58 943 
58 891 
58 838 
58 786 



58 734 
58 682 
58 630 
58 578 
58 526 



58 474 
58 422 
58 370 
58 319 
58267 



58 216 
58 164 
58 112 
58 061 
58 OIC 



57 958 
57 907 
57 856 
57 805 
57 753 



57 702 
57 651 
57 600 
57 549 
57 499 



57 448 
57 397 
57 346 
57 296 
57 245 



0.57 195 



Log, Tan. 



98 690 
98 687 
98 684 
98 681 
98 678 



98 674 
98 671 
98 668 
98 665 
98 662 



98 658 
98 655 
98 652 
98 649 
98 646 



98 642 
98 639 
98 636 
98 633 
98 630 



98 626 
98 623 
98 620 
98 617 
98 613 



98 610 
98 607 
98 604 
98 600 
98 597 



98 594 
98 591 
98 587 
98 584 
98 581 



98 578 
98 574 
98 571 
98 568 
98564 



98 561 
98 558 
98 554 
98 551 
98 548 



98 544 
98 541 
98 538 
98 534 
98 531 



98 528 
98 524 
98 52l 
98 518 
98 514 



98 5ll 
98 508 
98 504 
98 501 
98 498 



9. 98 494 



104° 



Log. Sin. 
606 



P. P. 



10 

20 

30 
40 
50 



54 

5.4 



53. 

5.3 
2 

71 

80 

8 
17 
26 
35 
44 



53 

5.3 
2 

9 
8 



52_ 

5.2 



2OI17 
30126 
40j35 
50143 



52 

5.2 



51_ 

5.1 
6 
6 
7 



51 

5.1 



50 

5.0 
5 
6 
7 



50i42 



50 

50 
5 
6 
7 
8 
16 



7 

6 

4 

8 

2 25 

6 33 

141 



49 

4 

5 

6 

7 

8 
16 
24 
33 
41 



49 

4-9 
7 
5 
3 
1 
3 
5 
6 



_ 16 
7|24 
0|32 
2140 





48 


48 


47 


47 




6 


4.8 


48 


4.7 


4.7 


7 


5 


6 


5 


6 


5 


5 


5 


5 


8 


6 


4 


6 


4 


6 


3 


6 


2 


9 


7 


3 


7 


2 


7 


1 


7 





10 


8 


1 


8 





7 


9 


7 


8 


20 


16 


1 


16 





15 


8 


15 


6 


30 


24 


2 


24 





23 


7 


23 


5 


40 


32 


3 


32 





31 


6 


31 


3 


50 


40 


4 


40 





39 


6 


39 


1 



0.3 








4 








4 








5 








6 





1 


1 


1 


1 


7 


1 


2 


3 


2 


2 


9 


2 



P. p. 



15^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



164'' 



Log. Sin. 



d. Log. Tan. c.d. Log. Cot. Log. Cos. 



P. P. 





1 
2 
3 
_4 
5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



9.41 299 
9.41 346 
9.41 394 
9.41441 
9-41 488 



9. 41 534 
9.41 581 
9-41 628 
9.41 675 
941 72] 



41 768 
9. 41 815 
9-41 861 
9. 41 908 
9. 41 954 



30 

21 
22 
23 
24 



25 
26 
27 
28 
29. 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 



9 42 690 
9. 42 735 
9. 42 781 
9-42 826 
9. 42 871 



42 000 
42 047 
42 093 
42 139 
42 185 



42 232 
42 278 
42 324 
42 369 
42 415 



9. 42 461 
9-42 507 
9-42 553 
9-42 598 
9-42 644 



42 917 

• 42 962 

• 43 007 

43 052 
■ 43 09? 



• 43 143 
43 188 

• 43 233 

• 43 278 

• 43 322 



43 367 
43 412 
43 457 
43 501 
• 43 546 



40 

41 
42 
43 
11 
45 
46 
47 
48 
49^ 

50 

51 
52 
53 
54^ 
55 
56 
57 
58 
59_ 
60 9 44 034 



43 591 
9.43 635 
43 680 
43 724 
43 768 



943 813 

43 857 

9. 43 901 

9-43 945 

43 989 



Log. Cos. 



9.42 805 
9.42 856 
9.42 906 
9-42 956 
9-43 007 



9.43 057 
9-43 107 
9.43 157 
9. 43 208 
9.43258 



9-44 299 
9-44 348 
9-44 397 
9 • 44 446 
9 . 44 49 



9-43 308 
9-43 358 
9- 43 408 
9-43 458 
9-43 508 



943 557 
9. 43 607 
9.43 657 
9-43 706 
9.43 756 



9. 43 806 
9. 43 855 
9.43 905 
9 43 954 
9 44 003 



9-44 053 
9-44 102 
9-44 151 
9-44 200 
9-44 249 



9-44 543 
9-44 592 
9.44 641 
9-44 690 
9-44 738 



9-44 787 
9-44 835 
9- 44 884 
9.44 932 
9. 44 981 



9.45 029 

9.45 077 

45 126 

45 174 

9-45 222 



9.45 270 
45 318 
45 367 
45 415 
45 463 



45 510 
45 558 

9-45 606 
45 654 

9-45 702 



9-45 749 



d. Log. Cot. c.d 



195 
144 
094 
043 
993 



942 
892 
842 
792 
742 



56 692 
56 642 
56 592 
56 542 
56 492 



56 442 
56 392 
56 343 
56 293 
56 243 



56 194 
56 144 
56 095 
56 045 
55 996 



55 947 
55 898 
55 848 
55 799 
55 750 



701 
652 
603 
554 
505 
456 
407 
359 
310 
261 



213 
164 
116 
067 
019 
970 
922 
874 
825 
777 



54 729 
54 681 
54 633 
54 585 
54 537 



54 489 
54 441 
54 393 
54 346 
54 298 



0-54 250 



Log. Tan, 



98 494 
98 491 
98 487 
98 484 
98 481 



98 477 
98 474 
98 470 
98 467 
98 464 



98 460 
98 457 
98 453 
98 450 
98 446 



98 443 
98 439 
98 436 
98 433 
98 429 



98 426 
98 422 
98 419 
98 415 
98 412 



98 408 
98 405 
98 401 
98 398 
98 394 



98 391 
98 387 
98 384 
98 380 
98 377 



98 373 
98 370 
98 366 
98 363 
98 359 




98 320 
98 316 
98 313 
98 309 
98 306 



98 302 
98 298 
98 295 
98 291 
98 288 



9 98 284 



-og. Sin. 



60 

59 
58 
57 
_56 
55 
54 
53 
52 
51 

50 

49 
48 
47 

ii 

45 
44 
43 
42 
ii 
40 
39 
38 
37 

35 
34 
33 
32 
^ 
30 
29 
28 
27 
26. 
25 
24 
23 
22 
21 
30 
19 
18 
17 
16 
15 
14 
13 
12 
IJL 
10 
9 



50_ 



5 





5 


5 


9 


5 


6 


7 


6 


7 


6 


7 


8 


4 


8 


16 


8 


16 


25 


2 


25 


33 


6 


33- 


42 


1 


41. 



50 


8 

6 
5 
3 
6 

3 
6 





49 


49 


48 


4? 


6 


4.9 


4.9 


4.8 


4 


7 


5 


8 


5 


7 


5.6 


5 


8 


6 


6 


6 


5 


6.4 


6 


9 


7 


4 


7 


3 


7.3 


7 


10 


8 


2 


8 


1 


8.1 


8 


20 


16 


5 


16 


3 


16.1 


16 


30 


24 


7 


24 


5 


24.2 


24 


40 


33 





32 




32.3 


32 


50 


41 


2 


40 


8 


40.4 


40 





47 


4" 


7 


46 


4( 


6 


4.7 


4.7 


4.6 


4 


7 


5 


5 


5 


5 


5 


4 


5 


8 


6 


3 


6 


2 


6 


2 


6 


9 


7 


1 


7 





7 





6 


10 


7 


9 


7 


g 


7 


7 


7 


20 


15 


8 


15 


6 


15 


5 


15 


30 


23 


7 


23 


5 


23 


2 


23 


40 


31 


G 


31 


3 


31 





30 


50 


39 


6 


39 


1 


38 


7 


38 





45 


45 


44 


4^ 


6 


4-5 


45 


4.4 


4 


7 


5 


3 


5 


2 


5 


2 


5 


8 


6 





6 





5 


9 


5 


9 


6 


8 


6 


7 


6 


7 


6 


10 


7 


6 


7 


5 


7 


4 


7 


20 


15 


1 


15 





14 


8 


14 


30 


22 


7 


22 


5 


22 


2 


22 


40 


30 


3 


30 





29 


6 


29. 


50 


37 


9 


37 


5 


37 


1 


36. 





4 


3 




6 


0.4 


0.3 





7 





4 





4 





8 





5 





4 





9 





6 





5 





10 





6 





6 





20 


1 


3 


1 


1 


1 


30 


2 





1 


7 


1. 


40 


2 




2 


3 


2 


50 


3 


3 


2 


9 


2- 



P.P. 



105° 



607 



74* 



TABLE VII.- 



16° 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



163** 



Log. Sin 



9.44 
44 



Log. 



034 

078 

122 

1 

209 

253 

297 

341 

384 

428 

472 

515 

559 

602 

646 

689 

732 

776 

819 

862 

905 

948 

991 

034 

077 

120 

163 

206 

249 

291 

334 

377 

419 

462 

504 

547 
589 
631 
674 
716 
758 
800 
842 
885 
92 

969 
Oil 
052 
094 
136 
178 
220 
261 
303 
345 
386 
428 
469 
511 
552 
593 
Cos 



Log. Tan. c.d. Log. Cot. Log. Cos 



45 749 
45 797 
45 845 
45 892 
45 940 



45 987 

46 035 
46 082 
46 129 
46 177 



46 224 
46 271 
46 318 
46 366 
46 413 



46 460 
46 507 
46 554 
46 601 
46 647 



46 694 
46 741 
46 788 
46 834 
46_8J1 
46 928 

46 974 

47 021 
47 067 
47 114 



47 160 
47 207 
47 253 
47 299 
47 345 



47 392 
47 438 
47 484 
47 530 
47 576 



47 622 
47 668 
47 714 
47 760 
47 806 



47 851 
47 897 
47 943 

47 989 

48 034 



48 080 
48 125 
48 171 
48 216 
48 262 



48 307 
48 353 
48 398 
48 443 
48 488 



48 534 
Log. Cot. 



54 250 
54 202 
54 155 
54 107 
54 060 



54 012 
53 965 
53 917 
53 870 
53 823 



53 776 
53 728 
53 681 
53 634 
53 587 



53 540 
53 493 
53 446 
53 399 
53 352 



53 305 
53 258 
53 212 
53 165 
53 118 



53 072 
53 025 
52 979 
52 932 
52 886 



52 839 
52 793 
52 747 
52 70g 
52 65 4 
52 608 
52 562 
52 516 
52 469 
52 423 



52 377 
52 331 
52 286 
52 240 
52 194 



52 148 
52 102 
52 057 
52 Oil 
5] 965 



5] 920 
51 874 
51 829 
51 783 
51 738 



51 692 
51 647 
51 602 
51 556 
5 1 511 
51 466 



Log. Tan 



98 284 
98 280 
98 277 
98 273 
98 269 



98 266 
98 262 
98 258 
98 255 
98 251 



98 247 
98 244 
98 240 
98 236 
98 233 



98 229 
98 225 
98 222 
98 218 
98 214 



98 211 
98 207 
98 203 
98 200 
98 196 



98 192 
98 188 
98 185 
98 181 
18 177 
98 173 
98 170 
98 166 
98 162 
98 158 
98 155 
98 151 
98 147 
98 143 
98 140 



98 136 
98 132 
98 128 
98 124 
98 121 



98 117 
98 113 
98 109 
98 105 
98J^02 
98 098 
98 094 
98 090 
98 086 
98 082 




106° 



Log. Sin. 
(308 



P. P. 



60 

59 
58 
57 
51 
55 
54 
53 
52 
11 
50 
49 
48 
47 
46 
45 
44 
43 
42 
-ii 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
■]4 
13 
12 
U_ 

10 

9 
8 
7 
6 
5 
4 
3 
2 

T 

o 

















48 47 47 1 


6 


4.8 


4 7 


4 7 1 


7 


5 


6 


5 


5 


5 


5 


8 


6 


4 


6 


3 


6 


2 


9 


7 


2 


7 


1 


7 





10 


8 





7 


9 


7 


8 


20 


16 





15 


8 


15 


6 


30 


24 





23 


7 


23 


5 


40 


32 





31 


6 


31 


3 


50 


40 





39 


6 


39 


' 



46_ 

4.6 
5 4 

■ 2 

•Q 

• 7 
•5 

• 2 

•Q 

• 7 



45 

4 

5 

6 

6 

7 
15 
22 
30 
37 



45 

4-5 
2 

7 
5 

5 

5 



6 

7 

8 

9 

10 

20 

30 

40 



44 

4 4 

5 

5 



50136 



43^ 

4.3 



43 

4.3 





42 


42 


41 


41 


6 


4 2 


4.2 


4.1 


4 


7 


4 


9 


4 


9 


4 


8 


4 


8 


5 


6 


5 


6 


5 


5 


5 


9 


6 


4 


6 


3 


6 


2 


6 


10 


7 


1 


7 





6 


9 


6 


20 


14 


] 


14 





13 


8 


13 


30 


21 


2 


21 





20 


7 


20 


40 


28 


3 


28 





27 


6 


27 


50 


35 


4 


35 





34 


6 


34 






4 








4 








5 








g 








6 





1 


3 


1 


2 





1 


2 


6 


2 


3 


3 


2 



P. p. 



73° 



17** 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



162** 



Log. Sin. d. Log, Tan. c.d. Log. Cot. Log. Cos.i 



39 



9.46 593 
9.46 635 
9.46 676 
9.46 717 
9.46 758 



9.46 799 
9.46 840 
9.46 881 
9.46 922 
9.46 963 



;t 9 



47 004 
9-47 045 
9-47 086 
9-47 127, ^, 
9-47 168 ^f 
9-47 2081 ^^ 
9.47 249 
9-47 290 
9.47 330l 
9473711 



y Q 



9. 47 411! 
9-47 452! 
9-47 492 
9-47 532, 
9.47 5731 



9.47 613! 
9-47 6531 
9.47 6941 

47 734; ' 
9.47 774! ' 



9.48 213 

48 252 

48 292 

9-48 331 

948 371 



48 410 

9-48 450 

48 489 

48 529 

948 568 



9-48 607 
9.48 646 
9.48 686 
9.48 725 
9.48 764 



48 803 
48 842 

9.48 881 
48 920 

9.48 959 



• 48 998 
Loc^. Cos. d, 



48 534 
-48 579 
-48 624 
-48 669 

48 714 



-48 984 

■ 49 028 
-49 073 
.49 118 
-49 162 



-49 207 
.49 252 

49 296 
.49 341 

49 385 



-49 430 
-49 474 
• 49 518 
-49 563 
-49 607 



• 49 651 
-49 695 
-49 740 
-49 784 

• 49 828 



-49 872 
■ 49 916 

• 49 960 

• 50 004 

• 50 048 



• 50 092 

• 50 136 

• 50 179 

• 50 223 

• 50 267 



• 50 311 
.50 354 

50 398 

• 50 442 
•50 485 



50 529 
•50 572 
•50 616 
•50 659 
-50 702 



• 50 746 
•50 789 

• 50 832 
50 876 

• 50 919 



• 50 962 

• 51 005 
-51 048 

• 51 091 

• 51 134 



•51 177 
Log. Cot. c. d. 



) 570 
) 525 
1481 
1437 
) 392 
) 348 
)304 
)260 
) 216 
) 172 



•49 689 

• 49 645 

• 49 602 

• 49 558 

• 49 514 



■ 49 471 

• 49 427 

• 49 384 

■ 49 340 

■ 49 297 



• 49 254 
•49 210 

• 49 167 
■49 124 

• 49 081 



49 038 
48 994 
48 951 
48 908 
48 865 



48 822 



• 98 059 

• 98 056 

• 98 052 

• 98 048 

• 98 044 



• 98 040 

• 98 036 

• 98 032 

• 98 028 

• 98 024 



• 98 021 

• 98 017 

• 98 013 

• 98 009 

• 98 005 



• 98 001 

• 97 997 

• 97 993 
.97 989 
.97 985 



.97 981 
.97 977 
.97 973 
.97 969 
• 97 966 



• 97 962 

• 97 958 
.97 954 

• 97 950 

• 97 946 



• 97 942 

• 97 938 

• 97 934 

• 97 930 

• 97 926 



.97 902 
.97 898 
.97 894 
.97 890 
.97 886 



.97 881 
.97 877 
.97 873 
.97 869 
.97 865 



.97 861 

• 97 857 

• 97 853 

• 97 849 

• 97 845 
.97 841 
.97 837 

• 97 833 
■ 97 829 

• 97 824 



97 820 



Log. Tan, Log. Sin, 



P. P. 





45 


45 


44 


44 


6 


4-5 


4-5 


4-4 


4-4 


7 


5 


3 


5 


2 


5 


o 


5 


1 


8 


6 





6 





5 


9 


5 


3 


9 


6 


8 


6 


7 


6 


7 


6 


6 


10 


7 


6 


7 


5 


7 


4 


7 


3 


20 


15 


1 


15 





14 


3 


14 


6 


30 


22 


7 


22 


5 


22 


2 


22 





40 


30 


3 


30 





29 


g 


29 


3 


50 


37 


9 


37 


5 


37 


1 


36 


6 



43 



4^3 


4 


5 


1 


5 


5 


8 


5 


6 


5 


6 


7 


2 


7 


14 


5 


14 


21 


7 


21 


29 





28 


36 


2 


35 



43 

3 

7 
4 
I 
3 
5 
6 
8 



4 


1 


41 


40 


4( 


6 4^I 


4^1 


4^0 


4 


7 4 


8 


4 


8 


4 


7 


4 


8 5 


5 


5 


4 


5 


4 


5 


9 6 


2 


6 


1 


6 


1 


6 


10 6 


9 


6 


8 


6 


7 


6 


20 13 


8 


13 


6 


13 


5 


13 


30 20 


7 


20 


5 


20 


2 


20 


40 27 


6 


27 


3 


27 





26 


50 34 


6 


34 


1 


33 


7 


33 



6 
7 
8 
9 

10 
20 13 
30 19 
40|26 
50i32 



7 
8 
9 
10 
20 
30 
40 
50 



39_ 

3^9i 
4 
5 
5 



2! 5 
9, 5 
6! 6 
1 13 
7 19 
3 26 
9 32 



38 

3^8 






40 


40 





sjo 


40 





6|0 


5 





70 


60 





710 


60 


1 


5jl 


3 1 


2 


2i2 


1 


3 


0|2 


62^ 


3 


7'3 


3 2^ 



4 3 

3 
4 
4 
5 
6 
1 
7 
3 



P. P. 



107° 



609 



73* 



18° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



161' 



' Log. Sin. d. Log. Tan. c. d. Log. Cot. Log. Cos. d. 



998 
037 
076 
114 
153 
192 
231 
269 
308 
346 
385 
423 
462 
500 
539 
577 
615 
653 
692 
730 
768 
806 
844 
882 
920 
958 
996 
034 
072 
110 
147 
185 
223 
260 
298 
336 
373 
411 
448 
486 
523 
561 
598 
635 
672 
710 
747 
784 
821 
858 

895 
932 
969 
006 
043 
080 
117 
154 
190 
227 
264 



Log, Cos. d 



53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
9-53 
Log. 



177 
220 
263 
306 
349 
392 
435 
477 
520 
563 
605 
648 
691 
733 
776 
818 
861 
903 
946 
988 
030 
073 
115 
157 
199 
241 
284 
326 
368 
410 
452 
494 
536 
578 
619 
661 
703 
745 
787 
828 
870 
912 
953 
995 
036 
078 
119 
161 
202 
244 
285 
326 
368 
409 
450 
49l 
533 
574 
615 
656 
697 
Cot. 



48 822 
48 779 
48 736 
48 693 
48 650 



608 
565 

522 
479 
437 



394 
351 
309 

266 
224 



181 
139 
096 
054 
012 



47 969 
47 927 
47 885 
47 842 
4 7 800 
47 758 
47 716 
47 674 
47 632 
47 590 



548 
506 
464 
422 
180 
338 
296 
255 
213 
171 



47 130 
47 088 
47 046 
47 005 
46 963 
46 922 
46 880 
46 839 
46 797 
46 756 



46 714 
46 673 
46 632 
46 591 
46 549 



46 508 
46 467 
46 426 
46 385 
46 344 
46 303 
Log, Tan. 



97 820 
97 816 
97 812 
97 808 
97 804 



97 800 
97 796 
97 792 
97 787 
97 783 
97 779 
97 775 
97 771 
97 767 
97 763 



97 758 
97 754 
97 750 
97 746 
97 742 



97 737 
97 733 
97 729 
97 725 
97 721 



97 716 
97 712 
97 708 
97 704 
97^00 
97 695 
97 691 
97 687 
97 683 
97 678 



97 674 
97 670 
97 666 
97 661 
97 657 



97 653 
97 649 
97 644 
97 640 
97 636 



97 632 
97 627 
97 623 
97 619 
97 614 
97 610 
97 606 
97 601 
97 597 
97 593 



97 588 
97 584 
97 580 
97 575 
97 571 

97 567 



108*" 



Log. Sin, 
610 



60 

59 
58 
57 
56 
55 
54 
53 
52 
11 
50 
49 
48 
47 

Ji 

45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 

30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 

li 

15 
14 
13 
12 
11 
10 
9 



P. P. 



43 43 



6 


4 


3 


4 


2 


4 


7 


5 





4 


9 


4 


8 


5 


■ 7 


5 


6 


5 


9 


6 


4 


6 


4 


6 


10 


7 


• 1 


7 


1 


7 


20 


14 


■ 3 


14 


1 


14 


30 


21 


• 5 


21 


o 


21 


40 


28 


• 6 


28 


3 


28 


50 


35-8 


35. 4l 


35 


41 41 


6 


4.1 


4.1 


7 


4 


• 8 


4 


8 


8 


5 


• 5 


5 


.4 


9 


6 


• 2 


6 


• 1 


10 


6 


■ 9 


6 


8 


20 


13 


■ 8 


13 


• 6 


30 


20 


•7 


20 


■ 5 


40 


27 


• 6 


27 


- 3 




50 


34 


6 


34 


1 



43 

2 
9 
6 
3 










39 


38 


38 


6 


3 9 


3.8 


3 8 


7 


4 


5 


4 


5 


4 


4 


8 


5 


2 


5 


1 


5 





9 


5 


3 


5 


8 


5 


7 


10 


6 


5 


^ 


4 


6 


3 


20 


13 





12 


8 


12 


6 


30 


19 


5 


19 


2 


19 





40 


26 





25 


6 


25 


3 


50 


32 


5 


32 


1 


31 


6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



37_ 

3-7 



37 

3-7 

4 

4 

5 

6 
12 
18 
24 
30 



36_ 

3-6 

4 

4 

5 

6 
12 
18 
24 
30 



6 





4 





7 





5 





8 





6 





9 





7 





10 





7 





20 


1 


5 


1 


30 


2 


2 


2 


40 


3 





2 


50 


3 


7 


3. 



P. p. 



71° 



19' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



160° 




60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
j46 
45 
44 
43 
42 
41 
40 
39 
38 
37 

35 
34 
33 
32 
II 
30 
29 
28 
27 
11 
25 
24 
23 
22 
IL 
20 
19 
18 
17 
11 
15 
14 
13 
12 
11 
10 



P. p. 



41 40 



4 


1 


4 





4. 


4 


8 


4 


7 


4. 


5 


4 


5 


4 


5- 


6 


1 


6 


1 


6. 


6 


8 


6 


7 


6. 


13 


6 


13 


5 


13. 


20 


5 


20 


2 


20 


27 


3 


27 





26. 


34 


1 


33 


7 


33. 



40 


6 
3 

6 
3 

6 
3 



39 



7 
8 
9 
10 
20 
30 
40 
50 



3 


9 


3 


4 


6 


4 


5 


2 


5 


5 


9 


5 


6 


6 


6 


13 


1 


13 


19 


7 


19 


26 


3 


26 


32 


9 


32 



39 

9 
5 
2 
8 
5 

5 

5 





37 


36 


3€ 


6 


3.7 


3.6 


3. 


7 


4 


3 


4 


2 


4. 


8 


4 


9 


4 


8 


4 


9 


5 




5 


5 


5. 


10 


6 


1 


6 


1 


6. 


20 


12 


3 


12 


1 


12. 


30 


18 


5 


18 


2 


18. 


40 


24 




24 


3 


24. 


50 


30 


8 


30 


4 


30. 





35 


35 


34 


6 


3.5 


3.5 


3 


7 


4 


1 


4 


1 


4 


8 


4 


7 


4 


g 


4 


9 


5 


3 


5 


2 


5 


10 


5 


9 


5 


8 


5 


20 


11 


8 


11 


6 


11. 


30 


17 


7 


17 


5 


17. 


40 


23 


6 


23 




23. 


50 


29 


6 


29 


1 


28. 



9 
10 
20 
30 
40 
50 



5 

0.5 









1 

2 

3 

4 



4 

0.4 



0. 

0. 

0. 

0. 
_ 1. 
22. 
02. _ 
7I3.3 



P. P. 



109° 



611 



70» 



TABLE VIl- 



20° 



-i^OGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



159° 



Log. Sin 





1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
li, 
15 
16 
17 
18 
19 



20 

21 

22 

23 

2A 

25 

26 

27 

28 

29_ 

30 ( 

31 

32 

33 

34 

35 
36 
37 
38 
39, 
40 
41 
42 
43 
44 



45 

46 

47 

48 

49_ 

50! 

51 

52 

53 

51 

55 

56 

57 

58 

59 

60 



53 405 
53 440 
53 474 
53 509 
53 544 
53 578 
53 613 
53 647 
53 682 
53 716 
53 750 
53 785 
53 819 
53 854 
53 888 
53 922 
53 956 

53 990 

54 025 
54 059 



54 093 
54 127 
54 161 
54 195 
54229 



54 263 
54 297 
54 331 
54 365 
54 398 



54 432 
54 466 
54 500 
54 534 
54 567 



54 601 
54 634 
54 668 
54 702 
54 735 



54 769 
54 802 
54 836 
54 869 
54 902 



54 936 

54 969 

55 002 
55 036 
55 069 



55 102 
55 135 
55 168 
55 202 
55 235 



55 268 
55 301 
55 334 
55 367 
55 400 
55 433 



Log. Cos 




60 

59 
58 
57 
^6_ 
55 
54 
53 
52 
II 
50 
49 
48 
47 
_46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
11 
30 
29 
28 
27 
16 
25 
24 
23 
22 
21 
20 
19 
18 
17 
JA 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 

5 
4 
3 
2 
1 
O 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



39 

39 



38 

38 

4 

5 

5 



12 



30il9 
40|25 
50!32 



4 

5 

5 

6 

_ 13 

7il9 

3126 

9132 



38 

38 



39 

3-9 



37 

37 





35 


34 


34 


6 


3-5 


3-4 


3-4 


7 


4 


1 


4 





3 


9 


8 


4 


6 


4 


6 


4 


5 


9 


5 


2 


5 


2 


5 


1 


10 


5 


8 


5 


7 


5 


6 


20 


11 


6 


11 


5 


11 


3 


30 


17 


5 


17 


2 


17 





40 


23 


3 


23 





22 


6 


50 


29 


1 


28 


7 


28 


3 



7 
8 
9 
10 
20 
30 
40 
50 



33 



33 

3 
8 
4 
9 
5 

5 

5 



5 4 

0.510.4 

60. 
6 0. 
70. 
8 0. 



3-3 


3 


3 


9 


3 


4 


4 


4 


5 





4 


5 


6 


5 


11 


1 


11 


16 


7 


16 


22 


3 


22 


27 


9 


27 



P. p. 



110° 



612 



69° 



TABLE VII 



-LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 158' 

P. P. 






38 


37 


3 


6 


38 


3.7 


3 


7 


4 


4 


4 


4 


4 


8 


5 





5 





4 


9 


5 


7 


5 


6 


5 


10 


6 


3 


6 


2 


6. 


20 


12 


6 


12 


5 


12- 


30 


19 





18 


7 


18- 


40 


25 


3 


25 





24. 


50 


31 


6 


31 


2 


30. 



10 
20 
30 
40 
50 



36 

3 



5 

1 
1 
2 
3 
30.4 



36 

36 

4 

4 
4 








33 

3 
3 

4 

4 

5 
11 
16 
22 



50I27 



33 

3.2 

3 

4 

4 

5 
10 
16 
21 
27 



32 

3.2 



31_ 



31 

31 





5 


5 


5 


6 


05 


0.5 





7 





6 





6 





8 





7 





6 





9 





8 





7 





10 





9 





8 





20 


1 


8 


1 


6 


1- 


30 


2 


7 


2 


5 


2. 


40 


3 


6 


3 


3 


3. 


50 


4 


6 


4 


1 


3. 



P. p. 



iir 



613 



68° 



22° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TAxNGENTS. 
AND COTANGENTS. 



157 



' Log. Sin. d. Log. Tan. c, d. Log. Cot. Log. Cos. d 



10 

n 

12 

13 

li 

15 

16 

17 

18 

ii 

20 

21 

22 

23 

21 

25 

26 

27 

28 

29_ 

30 

31 

32 

33 

34 9 

35 

36 

37 

38 

39. 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49L 

50 

51 

52 

53 

54 

55 

56 

57 

58 

51 

60 



57 357 
57 389 
57 420 
57 451 
57 482 



57 513 
57 544 
57 576 
57 607 
57 638 



57 669 
57 700 
57 731 
57 762 
57712 
57 823 
57 854 
57 885 
57 916 
57 947 



57 977 

58 008 
58 039 
58 070 
58 100 



58 131 
58 162 
58 192 
58 223 
58 253 



58 284 
58 314 
58 345 
58 375 
58406 



58 436 
58 466 
58 497 
58 527 
58 557 



58 587 
58 618 
58 648 
58 678 
58 708 



58 738 
58 769 
58 799 
58 829 
58 859 



58 889 
58 919 
58 949 

58 979 

59 009 
59 038 
59 068 
59 098 
59 128 
59 158 



59 188 
Log. Cos. 





• 61 364 

61 400 

9-61 436 

9. 61 472 

9. 61 507 



61 543 

61 579 

61 615 

61 651 

61 686 

9. 61 722 

61 758 

9. 61 794 

9. 61 829 

9-61 865 



61 901 
61 936 

61 972 

62 007 
62 043 



62 078 
9. 62 114 
9-62 149 
9. 62 185 

62 220 



9. 62 256 
62 291 
62 327 

9. 62 362 
62 397 



62 433 
62 468 
62 503 
62 539 
62 574 



9. 62 609 
9-62 644 
9. 62 679 
9. 62 715 
62 750 



9-62 785 
Log. Cot. 



36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

36 

35 

36 

36 

35 

36 

35 

36 

35 

36 

35 

35 

36 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 

35 



039 359 
039 322 
039 286 
0-39 250 
039 213 



0-39 177 
039 141 
039 105 
039 069 
039 032 



038 996 
038 960 
038 924 
038 888 
038 852 



038 816 
38 780 
038 744 
038 708 
0-38 672 



038 636 
038 600 
038 564 
038 528 
38492 



0-38 456 
038 420 
38 385 
38 349 
038 313 



38 277 
038 242 
038 206 
038 170 
038 135 



9. 96 716 
9-96 711 
9-96 706 
9.96 701 
9. 96 696 




9-96 639 
9.96 634 
9-96 629 
9. 96 624 
9-96 619 



96 613 
96 608 
96 603 
96 598 
• 96 593 



96 587 
96 582 
96 577 
96 572 
96 567 



038 099 
038 063 
038 028 
037 992 
37 957 



037 921 
037 886 
037 850 
037 815 
037 779 



037 
037 
037 
037 
037 



c.d, 



037 
0.37 
037 
037 
037 



744 
708 
673 
637 
602 
567 
531 
496 
461 
426 



037 390 
037 355 
0-37 320 
037 285 
0-37 250 
0-37 215 
Log. Tan. 



96 561 
96 556 
96 551 
96 546 
96 540 



9-96 535 
996 530 
996 525 
9-96 519 
9-96 514 



96 509 

• 96 503 
96 498 

• 96 493 

• 96 488 



96 482 
96 477 
96 472 
96 466 
96 461 



96 456 
96 450 
96 445 
96 440 
96 434 



96 429 
9 96 424 
9 96 418 
996 413 
9-96 408 



113° 



9. 96 402 
.og. Sin. 

614 



60 

59 
58 
57 

55 
54 
53 
52 
-51 
50 
49 
48 
47 
16 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
16 
25 
24 
23 
22 
21 
20 
19 
18 
17 
11 
15 
14 
13 
12 
11 
10 

9 

8 

7 
_6 

5 

4 

3 

2 
_1 




P. P. 



5030 



36 

36 



4 
4 
5 
5 
11 
17 
40,23 
5029 



10 
20 
30 



35_ 

5 
1 
7 
3 



35 

35 



4 
4 
5 
5 

8 11 
7117 
6i23 
6129 





31 


31 


6 


1-1 


31 


7 


3 


7 


3 


6 


8 


4 


2 


4 


1 


9 


4 


7 


4 


5 


10 


5 


2 


5 


1 


20 


10 


5 


10 


3 


30 


15 


7 


15 


5 


40 


21 





20 


6 


50 


26 


2 


25 


8 



403 
50 4 



5 5 

0-5 0-5 

6 0^6 

7 0-6 

8 0^7 

9 0.8 
8 16 
7 2-5 
6 3.3 
6 41 





30 


30 


29 


6 


3^0 


30 


2^9 


7 


3 


5 


3 


5 


3 


4 


8 


4 





4 





3 


9 


9 


4 


6 


4 


5 


4 


4 


10 


5 




5 





4 


9 


20 


10 


1 


10 





9 


8 


30 


15 


2 


15 





14 


7 


40 


20 


3 


20 





19 


g 


50 


25 


4 


25 





24 


6 



P. p. 



23° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 156« 



' Log. Sin, 



59 188 
59 217 
59 247 
59 277 
59 306 



59 336 
59 366 
59 395 
59 425 
59 454 



59 484 
59 513 
59 543 
59 572 
59 602 



59 631 
59 661 
59 690 
59 719 
59 749 



59 778 
59 807 
59 837 
59 866 
59 895 



59 924 
59 953 

59 982 

60 012 
60 041 




60 359 
60 388 
60 417 
60 445 
60 474 



60 503 
60 532 
60 560 
60 589 
60 618 



60 646 
60 675 
60 703 
60 732 
60 760 



60 789 
60 817 
60 846 
60 874 
60 903 



9 60 931 
Log. Cos. 



Log. Tan. c, d 



62 785 
62 820 
62 855 
62 890 
62 925 



62 960 

62 995 

63 030 
63 065 
63 100 



63 135 
63 170 
63 205 
63 240 
63 275 



63 310 
63 344 
63 379 
63 414 
63 449 



63 484 
63 518 
63 553 
63 588 
63 622 



63 657 
63 692 
63 726 
63 761 
63 795 



63 830 
63 864 
63 899 
63 933 
63 968 



64 002 
64 037 
64 071 
64 106 
64 140 



64 174 
64 209 
64 243 
64 277 
64312 



64 346 
64 380 
64 415 
64 449 
64 483 



64 517 
64 551 
64 585 
64 620 
64 654 



64 688 
64 722 
64 756 
64 790 
64 824 



64 858 
Log. Cot. 



Log, Cot, 



37 215 
37 179 
37 144 
37 109 
37074 



37 039 
37 004 
36 969 
36 934 
36 899 



36 864 
36 829 
36 794 
36 760 
36 725 
36 690 
36 655 
36 620 
36 585 
36 551 



516 
481 
447 
412 
377 
343 
308 
273 
239 
204 



36 170 
36 135 
36 101 
36 066 
36 032 



35 997 
35 963 
35 928 
35 894 
35 859 



35 825 
35 791 
35 756 
35 722 
35 688 



35 653 
35 619 
35 585 
35 551 
35 517 



35 482 
35 448 
35 414 
35 380 
35 346 



35 312 
35 278 
35 244 
35 209 
35 175 



0-35 141 
Log. Tan. 



Log. Cos. 



96 40S 
96 397 
96 392 
96 386 
96 38 



96 375 
96 370 
96 365 
96 359 
96 354 



96 349 
96 343 
96 338 
96 332 
96 327 
96 321 
96 316 
96 311 
96 305 
96 300 



96 294 
96 289 
96 283 
96 278 
96 272 



96 267 
96 261 
96 256 
96 251 
96 245 



96 240 
96 234 
96 229 
96 223 
96 218 



96 212 
96 206 
96 201 
96 195 
96 190 



96 184 
96 179 
96 173 
96 168 
96 162 



96 157 
96 151 
96 146 
96 140 
96 134 



96 129 
96 123 
96 118 
96 112 
96 106 



96 101 
96 095 
96 090 
96 084 
96 078 



9-96 073 



Log. Sin. 



60 

59 
58 
57 
56^ 
55 
54 
53 
52 
II 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
J6 
35 
34 
33 
32 

IL 

30 

29 
28 
27 
21 
25 
24 
23 
22 
21 
20 
19 
18 
17 

li 

15 

14 

13 

12 

11. 

10 

9 

8 

7 

_1 

5 

4 

3 

2 





P. P. 



7 
8 

9 
10 
20 
30 
40 
50 



35 



6 


3 


5 


3. 


7 


4 


1 


4 


8 


4 


7 


4. 


9 


5 


3 


5 


10 


5 


9 


5. 


20 


11 


8 


11 


30 


17 


7 


17 


40 


23 


g 


23 


50 


29 


6 


29 



34 



3 


4 


3 


4 





3 


4 


6 


4 


5 


2 


5 


5 


7 


5 


11 


5 


11 


17 


2 


17 


23 





22 


28 


7 


28 



35 

5 



34 

4 
9 
5 
1 
6 
3 

6 
3 



30 


5 

5 








29 29 



6 


3 


7 


3 


8 


4 


9 


4 


10 


5. 


20 


10. 


30 


15 


40 


20. 


50 


25. 



2 


9; 2 


9 


2 


3 


4 3 


4 


3 


3 


9 3 


8 


3 


4 


4 4 


3 


4 


4 


9 4 


8 


4 


9 


8 9 


6 


9 


14 


7|14 


5 


14 


19 


6119 


3 


19. 


24 


6 24 


1 


23 



6,0 

7,0 

80 

90 

10 1 

20 2 

30 3 

404 

50 5 



6 





5 





7 





5 





8 





7 





9 





8 











9 








1 


3 


1 





2 


7 


2 





3 


g 


3 





4 


6 


4 



28 
8 



P.P. 



113° 



615 



66° 



TABLE VII.- 



24° 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



155** 



O 

1 
2 
3 

_4 
5 
6 
7 
8 

_9^ 

10 

11 

12 

13 

li- 

15 

16 

17 

18 

19 



20 

21 

22 

23 

24 

25 

26 

27 

28 

2i 

30 

31 

32 

33 

3i 

35 

36 

37 

38 

39 



40 

41 

42 

43 

44 

45 

46 

47 

48 

4i 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 



Log. Sin. I d. Log. Tan. c.d. Log. Cot. Log. Cos. 



60 931 
60 959 

60 988 

61 016 
61 044 



61 073 
61 101 
61 129 
61 157 
6 1 186 

61 214 
61 242 
61 270 
61 298 
61 326 



61 354 
61 382 
61 410 
61 438 
61 466 



61 494 
61 522 
61 550 
61 578 
61 60J 
61 634 
61 661 
61 689 
61 717 
61_745 
61 772 
61 800 
61 828 
61 856 
6A_883 
61 911 
61 938 
61 966 

61 994 

62 021 



62 049 
62 073 
62 104 
62 131 
62 158 



62 186 
62 213 
62 241 
62 268 
62 295 



62 323 
62 350 
62 377 
62 404 
62 432 



62 459 
62 486 
62 513 
62 540 
62 567 



9-62 595 



Log. Cos 



64 858 
64 892 
64 926 
64 960 
64 994 



65 028 
65 062 
65 096 
65 129 
65 163 



65 197 
65 231 
65 265 
65 299 
65 332 



65 366 
65 400 
65 433 
65 467 
65 501 



65 535 
65 568 
65 602 
65 635 
6 5 669 
65 703 
65 736 
65 770 
65 803 
65 837 
65 870 
65 904 
65 937 

65 971 

66 00| 

66 037 
66 071 
66 104 
66 137 
66 171 



9-66 867 
Log. Cot, 



035 
35 



c.d 



141 
107 
073 
040 
006 
972 
938 
904 
870 
836 



34 802 
34 769 
34 735 
34 701 
34 667 



633 
600 
566 
532 
499 
46_ 
431 
398 
364 
331 



34 297 
34 263 
34 230 
34 196 
34 163 



129 
096 
062 
029 
9i§ 
96^ 
929 
895 
862 
829 



33 795 
33 762 
33 729 
33 696 
33 66| 
33 629 
33 596 
33 563 
33 529 
33 496 



33 463 
33 430 
33 397 
33 364 
33 331 



9 
9 
9 
9 
33 298J9 
33 265 
33 232 
33 198 
33 165 
33 132 



Log. Tan 



96 073 
96 067 
96 062 
96 056 
9 60 50 

96 045 
96 039 
96 033 
96 028 
96 022 
96 Oil 
96 011 
96 005 
95 999 
95^94 

95 988 
95 982 
95 977 
95 971 
95 965 



95 959 
95 954 
95 948 
95 942 
95 937 



95 931 
95 925 
95 919 
95 914 
95 908 



95 902 
95 896 
95 891 
95 885 
95 879 



95 873 
95 867 
95 862 
95 856 
95850 



95 844 
95 838 
95 833 
95 827 
95 821 



95 815 
95 809 
95 804 
95 798 
95 792 



95 786 
95 780 
95 774 
95 768 
95 763 



95 757 
95 751 
95 745 
95 739 
95 733 



995 727 



Log. Sin 



60 

59 
58 
57 
56_ 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
11 
25 
24 
23 
22 
21 
20 
19 
18 
17 

li 

15 
14 
13 
12 
11 
10 
9 



P. P. 





28 


28 


6 


2.8 


28 


7 


3 


3 


3 


2 


8 


3 


8 


3 


7 


9 


4 


3 


4 


2 


10 


4 


7 


4 


6 


20 


9 


5 


9 


3 


30 


14 


2 


14 





40 


19 





18 


6 


50 


23 


7 


23 


3 



27_ 



27 

7 
1 
6 

5 

5 

5 



6 5 

5 



2 


7 


2 


3 


2 


3 


3 


6 


3 


4 


1 


4 


4 


6 


4 


9 


1 


9 


13 


7 


13 


18 


3 


18 


122 


9 


22 



6 





60 


7 





70 


8 





80 


9 





90 


10 


1 


00 


20 


2 


01 


30 


3 


02 


40 


4 


03 


50 


5 


04 



P. p. 





34 


33 


33 


6 


3 4 


3.3 


3 3 1' 


7 


3 


9 


3 


9 


3 


8 i 


8 


4 


5 


4 


4 


4 


4 1 


9 


5 


1 


5 





4 


3 1 


10 


5 


6 


5 


6 


5 


5 1 


20 


11 


3 


11 


1 


11 


f 


30 


17 





16 


7 


16 


5 


40 


22 


6 


22 


3 


22 





50 


28 


3 


27 


9 


27 


5 



114° 



G16 



65° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



154° 



Log. Sin,| d. Log. Tan. c, d. Log. Cot. Lof?. Cos. d 



9-64 



924 
950 
976 
002 
028 
054 
080 
106 
132 
158 
184 



Log. Cos, 



■67 523 

• 67 556 

• 67 589 
•67 621 

67 654 

• 67 687 

• 67 719 
•67 752 
•67 784 
^6^817 

• 67 849 

• 67 882 

• 67 914 
■67 947 

• 67 979 



• 68 012 

• 68 044 

• 68 077 
■ 68 109 

• 68 141 



• 68 174 

• 68 206 
•68 238 
•68 271 
•68 303 



■ 6.8 335 

■ 68 368 
. 68 400 

■ 68 432 
• 68 464 



• 68 497 

• 68 529 

• 68 561 

• 68 593 

• 68 625 



9^68 657 
9. 68 690 
9-68 722 
9 •68 754 
9^68 786 
968 818 
Log. Cot. 



c.d, 



33 132 
33 100 
33 067 
33 034 
_33_001 
32 968 
32 935 
32 902 
32 869 
32 836 
32 803 
32 771 
32 738 
32 705 
32 672 



32 640 
32 607 
32 574 
32 541 
•32 509 



32 476 
32 443 
32 411 
32 378 
32 345 



32 313 
32 280 
32 248 
32 215 
32 183 




826 
793 
761 
729 
696 



632 
600 

567 
535 



31 503 
31 471 
31 439 
31 406 
31 374 



31 342 
31 310 
31 278 
31 246 
31 214 



31 182 



Log, Tan, 




95 727 
95 721 
95 716 
bo 710 
m 704 



95 698 
95 692 
95 686 
95 680 
95 674 



95 688 
95 662 
95 656 
95 650 
95 644 



95 638 
95 632 
95 627 
95 621 
95 615 



95 549 
95 543 
95 537 
95 530 
95 524 



95 518 
95 512 
95 506 
95 500 
95 494 



95 488 
95 482 
95 476 
95 470 
95 464 



95 458 
95 452 
95 445 
95 439 
95 433 



95 427 
95 421 
95 415 
95 409 
95 403 



95 397 
95 390 
95 384 
95 378 
95 372 
95 366 



115*' 



Log. Sin, 
617 



P. P. 





33 


33 


32 


6 


3-3 


3-2 


3^2 


7 


3 


8 


3 


8 


3 


7 


8 


4 


4 


4 


3 


4 


2 


9 


4 


9 


4 


9 


4 


8 


10 


5 


5 


5 


4 


5 


3 


2C 


11 





10 


8 


10 


g 


3G 


16 


5 


16 


2 


16 





40 


22 





21 


6 


21 


3 


50 


27 


5 


27 


1 


26 


6 



27 

7 
1 
6 

5 

5 

5 



26 26 



6 


2 


7 


3 


8 


3 


9 


4 


10 


4 


20 


9 


30 


13 


40 


18 


50 


22 



2^6 


2^6 


2 


3 


1 


3 





3 


3 


5 


3 


4 


3 


4 





3 


9 


3 


4 


4 


4 


3 


4 


8 


8 


8 


6 


8 


]3 


2 


13 





12 


17 


6 


17 


3 


17 


22 


1 


21 


6 


21 



i 


» 


6 


/i 


06 


06 








7 





7 








8 





8 





1 








9 





1 


1 


1 








o 


1 


2 





1 


3 


2 


3 





2 


4 


3 


4 





3 


5 


4 


5 





4 



25_ 

5 

4 
8 
2 
5 
7 

2 



P. P. 



64' 



26** 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS, 
AND COTANGENTS. 



153 



Log. Sin. d. Log, Tan. c. d, Log. Cot, Log. Cos 



64 184 
64 210 
64 236 
64 262 
64 287 



64 313 
64 339 
64 365 
64 391 
64 416 



64 442 
64 468 
64 493 
64 519 
64 545 



64 570 
64 596 
64 622 
64 647 
64 673 



64 698 
64 724 
64 749 
64 775 
64 800 



64 826 
64 851 
64 876 
64 902 
64 927 



64 952 

64 978 

65 003 
65 028 
65 054 



65 079 
65 104 
65 129 
65 155 
65 180 



65 205 
65 230 
65 255 
65 280 
65 305 



65 331 
65 356 
65 381 
65 406 
65 431 



65 456 
65 481 
65 506 
65 530 
65 555 



65 580 
65 605 
65 630 
65 655 
65 680 



9-65 704 



Log. Cos. 



68 818 
68 850 
68 882 
68 914 
68 946 



68 978 

69 010 
69 042 
69 074 
69 106 



69 138 
69 170 
69 202 
69 234 
69 265 



69 297 
69 329 
69 361 
69 393 
69 425 



69 456 
69 488 
69 520 
69 552 
69 583 
69 615 
69 647 
69 678 
69 710 
69 742 



69 773 
69 805 
69 837 
69 868 
69 900 



69 931 
69 963 

69 994 

70 026 
70 058 



70 403 
70 435 
70 466 
70 497 
70 529 



70 560 
70 59l 
70 623 
70 654 
70 685 



70 716 



Log. Cot. c. d 



0.30 226 
0-30 194 
0.30 163 
0-30 131 
0-30 100 



0-30 543 
0.30 511 
0.30 480 
0.30 448 
0.30 416 



0.30 384 
0.30 353 
0.30 321 
0.30 289 
0.30 258 



0.30 068 
0.30 037 
0.30 005 
0.29 973 
0.29 942 



0.29 
0.29 
0.29 
0.29 
0.29 



0.29 
0.29 
0.29 
0.29 
0.29 
0.29 
0.29 
0.29 
0.29 
p. 29. 
029 
029 
0.29 



910 
879 
847 
816 
785 
753 
722 
690 
659 
628 
596 
565 
533 
502 
471 
439 
408 
377 
346 
314 



0.29 283 
Log. Tan, 



95 366 
95 360 
95 353 
95 347 
95 341 




95 273 
95 267 
95 260 
95 254 
95 248 



95 242 
95 235 
95 229 
95 223 
95 217 



95 210 
95 204 
95 198 
95 191 
95 185 



95 179 
95 173 
95 166 
95 160 
95 154 

95 147 
95 141 
95 135 
95 128 

9512:? 



95 116 
95 109 
95 103 
95 097 
95 090 



95 084 
95 078 
95 071 
95 065 
95_0_58 
95 052 
95 046 
95 039 
95 033 
95 026 



95 020 
95 014 
95 007 
95 001 
94 994 



9-94 988 

Log. Sin. 



P.P. 



7 
8 
9 
10 
20 
30 
40 
50 



32 



3 


2 


3 


3 


8 


3 


4 


3 


4 


4 


9 


4 


5 


4 


5 


10 


8 


10 


16 


2 


16 


21 


6 


21 


27 


1 


26. 



31 



3 


I 


3 


7 


4 


2 


4 


7 


5 


2 


10 


5 


15 


V 


21 





26 


2 



33 

2 
7 
2 
8 
3 
6 

g 

3 
6 



31 

3.1 
3.6 

4.1 
4-6 
5.1 
10-3 
15-5 
20.6 
25-8 





36 


35 


2i 


6 


2.6 


2.5 


2 


7 


3 


d 


3 





2 


8 


3 


4 


3 


4 


3 


9 


3 


9 


3 


8 


3 


10 


4 




4 


2 


4 


20 


8 


6 


8 


5 


8 


30 


13 





12 


7 


12 


40 


17 


3 


17 





16 


50 


21 


6 


21 


2 


20 





2? 


6 




6 


6 


2.4 


0.6 





7 


2 


8 





7 





8 


3 


2 





8 


9 


9 


3 


7 


1 








10 


4 


1 


1 


1 


1 


20 


8 


1 


2 


1 


2 


30 


12 


2 


3 


2 


3 


40 


16 


3 


4 


3 


4 


50 


20 


4 


5 


4 


5. 



P.P. 



116** 



618 



63° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



152* 



Log. Sin, d. Log. Tan. c. d. Log. Cot Log. Cos 



9-65 
65 
65 
65 
65 



66 



704 
729 
754 
779 
803 
828 
853 
878 
902 
927 
95l; 
976 
001! 
025 
MO I 
074' 
099 
123 
148 
172 

197. 
221 
246! 
270, 
294 1 
319| 
343' 
367! 
392! 
416; 
440 1 
4651 
489; 
513! 
537: 
561 
586 
610! 
634 
_658 | 
682 
708 
730 
754 
778 
802 
826 
850 
874 



66 



67 
67 
67 
67 
67 
67 
Log. 



922 
946 
970 
994 
018 
042 
066 
089 
113 
137 
161 
Cos 



25 
24 
25 
24 
25 
24 
25 
24 
j 24 

24 
25 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 
24 
24 
24 
23 
24 

24 
24 
23 
24 
23 
24 



70 716 
70 748 
70 779 
70 810 
70 841 



70 872 
70 903 
70 935 
70 966 
70 997 



71 028 
71 059 
71 090 
71 121 
71 152 



71 183 
71 214 
71 245 
71 276 
71 307 



71 338 
71 369 
71 400 
71 431 
71 462 



71 493 
71 524 
71 555 
71 586 
71 617 



71 647 
71 678 
71 709 
71 740 
71 771 



71 801 
71 832 
71 863 
71 894 
71 925 




72 262 
72 292 
72 323 
72 354 
72 384 



72 415 
72 445 
72 476 
72 506 
72 537 



72567 



Log. Cot' 



29 283 
29 252 
29 221 
29 190 
29 158 



29 127 
29 096 
29 065 
29 034 
29 003 
28 972 
28 940 
28 909 
28 878 
28 847 



28 816 
23 785 
28 754 
28 723 
28 692 



28 661 
28 630 
28 599 
28 568 
28 537 



508 
476 
445 
414 
383 
352 
321 
290 
260 
229 



28 198 
28 167 
28 138 
28 106 
28 075 



28 044 
28 014 
27 983 
27 952 
27 921 



27 891 
27 860 
27 830 
27 799 
27 768 



27 738 
27 707 
27 677 
27 646 
27 615 



27 585 
27 554 
27 524 
27 493 
27 463 



27 432 



Log. Tan 



94 988 
94 981 
94 975 
94 969 
94 962 



94 956 
94 949 
94 943 
94 936 
94 930 



94 923 
94 917 
94 910 
94 904 
94 897 



94 891 
94 884 
94 878 
94 871 
94 865 



94 858 
94 852 
94 845 
94 839 
94 832 




94 727 
94 720 
94 713 
94 707 
94 700 



94 693 
94 687 
94 680 
94 674 
94 667 



9-94 593 
Log. Sin. 



60 

59 
58 

57 

55 
54 
53 
52 
11 
50 
49 
48 
47 
J8 
45 
44 
43 
42 
41 
40 
39 
38 
37 
11 
35 
34 
33 
32 
11 
30 
29 
28 
27 
26_ 

25 
24 
23 
22 
21 

20 

19 
18 
17 
11_ 
15 
14 
13 
12 
11 
10 



P. P. 



6 
7 
8 

9 
10 
20 10 
30|15 
40121 
50i26 



31 

3 " 
3 

4 
4 
5 



31 

3.11 



30 

3 



3 10 
5 15 
6!20 
8l25 



7 
8 
9 
10 
20 
30 
40 
50 



35 

2.5 

2.9 

3.3 

3.7 

4.1 

8.3 

12.5 

16-6 

20. § 



24: 

2.4! 



10 

20 

30 

40^16 

50120 



8! 2 

2| 3 

7 3 

1 4 

l! 8 

2:12 

3 16 
420 



34 

4 
8 

2 
6 





35 

2.3 



2 
3 
3 
3 

7 
Olll. 
15- 
0119. 







7 


6 


6 


0-7 


0-6! 


7 





8 





7 


8 





9 





g 


9 


1 





1 





10 


1 


1 


1 


1 


20 


2 


3 


2 


1 


3013 


5 


3 


2 


4014 


g 


4 


3 


50 


5 


8 


5 


4 



P. p. 



ll?** 



()]9 



63' 



28* 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, 

AND COTANGENTS. 1; 



' Log. Sin 



67 161 
67 184 
67 208 
67 232 
67 256 



67 279 
67 303 
67 327 
67 350 
67374 



67 397 
67 421 
67 445 
67 468 
67 492 



67 515 
67 539 
67 562 
67 586 
67 609 



67 633 
67 656 
67 679 
67 703 
67 726 



67 750 
67 773 
67 796 
67 819 
67 843 



67 866 
67 889 
67 913 
67 936 
67 959 



67 982 

68 005 
68 029 
68 052 
68 075 



68 098 
68 121 
68 144 
68 167 
68 190 



68 213 
68 236 
68 259 
68 282 
68 305 



9-68 557 
Log. Cos. 



Log. Tan.|c.d.:Log. Cot. Log. Cos 



72 567 
72 598 
72 628 
72 659 
72 689 



72 719 
72 750 
72 780 
72 811 
72 841 



72 871 
72 902 
72 932 
72 962 
72 993 



73 023 
73 053 
7S 084 
73 114 
73 144 



73 174 
73 205 
73 235 
73 265 
73 295 



73 325 
73 356 
73 386 
73 416 
73 446 




73 777 
73 807 
73 837 
73 867 
73 897 



73 927 
73 957 

73 987 

74 017 
74 047 



74 076 
74 106 
74 136 
74 166 
74 196 



74 226 
74 256 
74 286 
74 315 
74 345 



9-74 375 
Log. Cot. 



0-27 432 
0.27 402 
0-27 371 
0-27 341 
0-27 311 



0-27 
0-27 
0-27 
0-27 
!0-27 



280 9. 
250 9. 



9.94 593 
9.94 587 
94 580 
9.94 573 
9.94 566 



0-27 
0-27 
0-27 
0-27 
0-27 



219 
189 

133 

128 
098 
067 
037 
007 



0-26 976 
0-26 946 
0-26 916 
0.26 886 
0-26 855 



0.26 82e 
0.26 7 
0.26 765 
0.26 734 
0.26 704 



0.26 674 
0.26 644 
0.26 614 
0.26 584 
0.26 553 



0-26 52S 
0.26 493 
0.26 463 
0.26 433 
0.26 403 



26 373 
26 343 
26 313 
26 283 
26 253 



0-26 
0.26 
0.26 
0.26 
0.26 



0.26 
0.26 
0.26 
0.25 
0.25 



223 
193 
163 
133 
103 
073 
043 
013 
983 
953 



c.d. 



25 92? 

25 89S 
0.25 86S 
0.25 833 
0.25 8C4 



0-25 774 
0.25 744 
0.25 714 
0.26 684 
0.25 654 



94 560 
94 553 
94 546 
94 539 
94 533 



94 526 
94 519 
94 512 
94 506 
.94499 

9-94 492 
9.94 485 
9.94 478 
9.94 472 
94 465 



9. 94 458 
9.94 451 
9 . 94 444 
9.94 437 
9.94 431 



9.94 424 
9.94 417 
9.94 410 
9.94 403 
9.94 396 



94 390 
94 383 
94 376 
94 369 
. 94 362 
94 355 
. 94 348 
• 94 341 
94 335 
94 328 



.94 321 
.94 314 
.94 307 
.94 300 
• 94293 



94 286 
94 279 
94 272 
94 265 
94 258 



94 251 
94 245 
94 238 
94 231 
94 224 



0-25 625 



9.94 217 
9.94 210 
9.94 203 
9.94 196 
94 189 
9.94 182 



Log. Tan. 



118** 



Log. Sin. 
020 



60 

59 
58 
57 
56, 
55 
54 
53 
52 
51 

50 

49 
48 

47 

ii 

45 
44 
43 
42 
ii, 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
2i 
25 
24 
23 
22 
_21_ 
20 
19 
18 
17 

15 
14 
13 
12 

n 

10 



p. p. 





30 


r.o 


29„ 




6 


3.0 


a.o 


2-9 




7 


3 


5 


3.5 


3 


4 




8 


4 





4.0 


3 


9 




9 


4 


6 


4.5 


4 


d 




10 


5 


1 


5.0 


4 


9 




20 


10 




10.0 


9 


8 




30 


15 


2 


15.0 


14 


7 




40 


20 





20.0 


19 


6 




50 


25 


4 


25.0 


24 


6 







24 


6 


2.4 


7 


2.8 


8 


3.2 


9 


3.6 


10 


4.0 


20 


8.0 


30 


12.0 


40 


16.0 


50 


20.0 





23 


23 


22 


6 


2.3 


2-3 


2.2 


7 


2 


7 


2 


7 


2 


6 


8 


3 


1 


3 





3 





9 


3 


5 


3 


4 


3 


4 


10 


3 


9 


3 


3 


3 


7 


20 


7 


8 


7 


6 


7 


5 


30 


11 


7 


11 


5 


11 


2 


40 


15 


6 


15 




15 





50 


19 


6 


19 


1 


18 


7 





7 




6 


6 


0.7 


0.6 


7 





8 





7 


8 





9 





s 


9 


1 





i 





10 


1 


1 


1 


1 


20 


2 


3 


2 


1 


30 


3 


5 


3 


2 


40 


4 


6 


4 


3 


50 


5 


8 


5 


4 



P.P. 



61" 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS, 
AND COTANGENTS. 



ISO* 



Log. Sin 




68 897 

68 920 
68 942 
68 965 
68 987 



69 010 
69 032 
69 055 
69 077 
69099 



69 122 
69 144 
69 167 
69 189 
69 211 



69 234 
69 256 
69 278 
69 301 
69 323 



69 345 
69 367 
69 390 
69 412 
69 434 



69 456 
69 478 
69 500 
69 523 
69 545 



69 567 
69 589 
69 611 
69 633 
69 655 



69 677 
69 699 
69 721 
69 743 
69765 



69 787 
69 809 
69 831 
69 853 
69 875 



9.69 897 
Log. Cos. 



d. Log. Tan. c.d. Log, Cot. Log. Cos, 



9.74 821 
9.74 850 
9.74 880 
9.74 909 
9.74 939 



9.74 376 
9-74 405 
9-74 435 
9-74 464 
9-74 494 



9.74 524 
9.74 554 
9.74 583 
9.74 613 
9-74 643 



9.74 672 
9.74 702 
9.74 732 
9.74 761 
9.74 791 



9.74 969 

9.74 998 

9.75 028 
9.75 057 
9.75 087 



9.75 116 
9.75 146 
9.75 175 
9.75 205 
9.75 234 



9.75 264 
9.75 293 
9.75 323 
9.75 352 
9-75 382 



9.75 411 
9.75 441 
9.75 470 
9.75 499 
9-75 529 



9.75 558 
9-75 588 
9-75 617 
9.75 646 
9-75 676 



9.75 705 
9-75 734 
9-75 764 
9.75 793 
9-75 822 



9.75 851 
9.75 881 
9-75 910 
9-75 939 
9-75 968 



9-75 998 
9-76 027 
9-76 056 
9-76 085 
9-76 115 



9-76 144 
Log. Cot, 



30 
30 
29 
30 

29 
30 
29 
29 
30 
29 
30 
29 
29 
30 
29 
29 
29 
29 
30 

29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 
29 
29 
29 
29 
29 

29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 

—, 



0-25 625 
0-25 595 
0-25 565 
0-25 535 
0-2 5 505 
0-25 476 
0-25 446 
0-25 416 
0-25 387 
0-25 357 



0-25 327 
0-25 297 
0-25 268 
0-25 238 
0-25 208 



0-25 17? 
0.25 149 
0-25 120 
0-25 090 
0-25 060 



0-25 031 
0-25 001 
0.24 972 
0-24 942 
0-24 913 



0.24 883 
0-24 854 
0-24 824 
0-24 795 
0-24 765 



0-24 736 
0-24 706 
0-24 677 
0.24 647 
0^24 618 
0. 
0. 
0. 



24 588 
24 55? 
24 52? 
0.24 500 
0-24 471 



0.24 441 
0-24 412 
0.24 383 
0.24 353 
0.24 324 



0.24 295 
0.24 265 
0.24 236 
0.24 207 
0.24 177 



0.24 148 
0.24 119 
0.24 090 
0.24 060 
0.24 031 



0.24 002 
0.23 973 
0.23 943 
0.23 914 
023 885 



0.23 856 
Log, Tan. 



94 182 
94 175 
94 168 
94 161 
94 154 



94 147 
94 140 
94 133 
94 126 
94 118 



94 111 
94 104 
94 097 
94 090 
94 083 



94 076 
94 069 
94 062 
94 055 
94 048 



94 041 
94 034 
94 026 
94 01? 
94 012 



93 934 
93 926 
93 91? 
93 912 
93 905 



93 898 
93 891 
93 883 
93 876 
93 869 



93 862 
93 854 
93 847 
93 840 
93 833 



93 826 
93 818 
93 811 
93 804 
93 796 



93 789 
93 782 
93 775 
93 767 
93 760 



9.93 753 
Log, Sin. 



60 

59 
58 
57 
56_ 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 

25 
24 
23 
22 
_2_1 

20 
19 
18 
17 
16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 
4 
3 
2 
1 




P. P. 





30 


31 


6 


3-0 


2 


7 


8.5 


3 


8 


4.0 


3 


9 


45 


4 


10 


5.0 


4 


20 


10.0 


9 


30 


15.0 


14 


40 


20.0 


19 


50 


25-0 


24 



3d 

29 



7 2 

8 3 

9 3 
10, 3 
20 7 
30 11 
40 15 
50ll9 



7 14 
6il9 
6l24 





33 


33 


3 


6 


2.2 


2.2 


2 


7 


2 


6 


2 


5 


2 


8 


3 





2 


9 


2 


9 


3 


4 


3 


3 


3 


10 


3 


7 


3 


f) 


3 


20 


7 


5 


7 


3 


7. 


30 


11 


2 


11 





10. 


40 


15 





14 


(^ 


14. 


50 


18 


7 


18 


3 


17. 





7 


7 


6 


0.7 


07 


7 


0.9 


08 


8 


1.0 


09 


9 


1.1 


i.d 


10 


1.2 


l.T 


20 


2.5 


2..'^ 


30 


3.7 


35 


40 


5.0 


4.6 


50 


6-2 


5-8 



P. p» 



621 



60* 



30° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



149** 



Log. Sin, 



9 



25 f9 

26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



897 
919 
940 
962 
981 
006 
028 
050 
071 
093 
115 
137 
158 
180 
202 
223 
245 
267 
288 
310 

331 
353 
375 
396 
418 
439 
461 
482 
504 
525 

547 
568 
590 
611 
632 

654 
675 
696 
718 
739 



Log, Tan. c. d. Log, Cot. Log. Cos. d. 



71 



760 
782 
803 
824 
846 
867 
888 
909 
930 
952 

973 
994 
015 
036 
057 
078 
099 
121 
142 
163 

m. 



76 144 
76 173 
76 202 
76 231 
76 260 



76 289 
76 319 
76 348 
76 377 
76 406 



76 435 
76 464 
76 493 
76 522 
76 551 



76 580 
76 609 
76 638 
76 667 
76 696 



76 725 
76 754 
76 783 
76 812 
76 841 



76 870 
76 899 
76 928 
76 957 
76 986 



77 015 
77 043 
77 072 
77 101 
77 130 



77 159 
77 188 
77 217 
77 245 
77 274 



77 303 
77 332 
77 361 
77 389 
77 418 



77 447 
77 476 
77 504 
77 533 
7 7 562 
77 591 
77 619 
77 648 
77 677 
77 705 



77 734 
77 763 
77 791 
77 820 
77 849 



77 877 



r 

i 





23 856 
23 827 
23 797 
23 768 
23 739 



9.93 753 
9.93 746 
9.93 738 
9.93 731 
9.93 724 



23 710 
23 681 
23 652 
23 623 
23 594 



23 565 
23 535 
23 506 
23 477 
23 448 



419 
390 
361 
332 
303 



274 
245 
216 
187 
158 9 



93 716 
9.93 709 
9-93 702 
9.93 694 
9-93 687 



9-93 680 
9.93 672 
9.93 665 
9.93 658 
9-93 650 



93 643 
93 635 
93 628 
93 621 
93 613 



23 129 
23 101 
23 072 
23 043 
23 014 



22 985 
22 956 
22 927 
22 898 
22 869 



22 841 
22 812 
22 783 
22 754 
22 725 



22 696 
22 668 
22 639 
22 610 
22 581 



22 553 
22 524 
22 495 
22 466 
22 438 



22 409 
22 380 
22 352 
22 323 
22 294 



22 266 
22 237 
22 208 
22 180 
22 151 



22 122 



93 606 
93 599 
93 591 
93 584 
93 576 




9. 93 495 
9-93 487 
9-93 480 
9 . 93 472 
9 93 465 



9-93 457 
9-93 450 
9 . 93 442 
9.93 435 
9 93 427 



9. 93 420 
9. 93 412 
9.93 405 
9.93 397 
9-93 390 



9. 93 382 
9-93 374 
9.93 367 
9. 93 359 
9-93 352 



9.93 344 
9.93 337 
9.93 329 
9.93 321 
9.93 314 



9 93 306 



60 

59 
58 
57 
11 
55 
54 
53 
52 
II 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 



P.P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



29 

2.9 

3-4 

3.9 

4.4 

4.9 

9.8 

14.7 

19.6 

24.6 



29 

2 

3 

3 

4 

4 

9 
14 
19 
24 



28 
2.8 
3 
8 
3 
7 
5 
2 

7 





22 


2T 


2 


6 


2.2 


2.1 


2. 


7 


2 


5 


2.5 


2. 


8 


2 


9 


2-8 


2. 


9 


3 


3 


3.2 


3. 


10 


3 


6 


3.6 


3. 


20 


7 


3 


7.1 


7. 


30 


11 





10.7 


10. 


40 


14 


g 


14.3 


14. 


50 


18 


3 


17.9 


17- 





8 


; 


7 


F 


6 


0.8 


0.7 





7 





9 


0.9 





8 


1 





1.0 





9 


1 


2 


1.1 


1 


10 


1 


3 


1.:^ 


1 


20 


2 


5 


2.5 


2 


30 


4 





3-7 


3 


40 


5 


3 


5.0 


4 


50 


6 


6 


6.2 


5 



Log, Cos, d 



Log. Cot. c.d. Log, Tan. 



Log. Sin. 



P.P. 



120' 



622 



69' 



TABLE Vir.- 



31° 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



148° 



Log. Sin 



9 




71 184 
71 205 
71 226 
71247 
71 268 



71498 
71 518 
71 539 
71 560 
71 581 



71 601 
71 622 
71 643 
71 664 
71 684 



71 705 
71 726 
71 746 
71 767 
71 788 



71 808 
71 829 
71 849 
71 870 
71 891 



71 911 
71 932 
71 952 
71 973 
71 993 



72 014 
72 034 
72 055 
72 075 
72 096 



72 116 
72 136 
72 157 
72 177 
72 198 



72 218 
72 238 
72 259 
72 279 
72 299 



72 319 
72 340 
72 360 
72 380 
72 4«0 



72 421 
Log. Cos. 



Log. Tan, c.d. Log. Cot. Log. Cos 




79 



877 
906 
934 
963 
992 
020 
049 
077 
106 
134 
163 
19l 
220 
248 
277 

305 

334 

362 

391 

41 

448 

476 

505 

533 

56l 

590 
618 
647 
675 
703 
732 
760 
788 
817 
845 
873 
902 
930 
958 
987 
015 
043 
071 
100 
128 
156 
184 
213 
241 
269 

297 
325 
354 
382 
410 
438 
466 
494 
522 
551 
579 



Log. Cot 



22 122 
22 094 
22 065 
22 037 
22 008 



21 979 
21 951 
21 922 
21 894 
21 865 



837 
808 
780 
751 
723 
694 
666 
637 
609 
580 



21 552 
21 523 
21 495 
21 467 
21 438 



21 410 
21 381 
21 353 
21 325 
21 296 



21 268 
21 239 
21 211 
21 183 
21 154 



21 126 
21 098 
21 070 
21 041 
21 013 



20 985 
20 956 
20 928 
20 900 
20 872 



20 843 
20 815 
20 787 
20 759 
20 731 



20 702 
20 674 
20 646 
20 618 
20 590 



20 561 
20 533 
20 505 
20 477 
20 449 



0-20 421 
Log. Tan. 




93 306 
93 299 
93 291 
93 284 
93 276 



93 192 
93 184 
93 177 
93 169 
93 16l 



93 153 
93 146 
93 138 
93 130 
93 123 



93 115 
93 107 
93 100 
93 092 
93 084 



93 076 
93 069 
93 061 
93 053 
93 045 



93 038 
93 030 
93 022 
93 014 
93 006 



92 999 
92 991 
92 983 
92 975 
92 967 



92 960 
92 952 
92 944 
92 936 
92 928 



92 920 
92 913 
92 905 
92 897 
92 889 



92 881 
92 873 
92 865 
92 858 
92 850 



9-92 842 
Log. Sin. 



P. P. 





29 


28 


6 


2.9 


2-81 


7 


3 


4 


3 


3 


8 


3 


8 


3 


8 


9 


4 


3 


4 


3 


10 


4 


8 


4 


7 


20 


9 


6 


9 


5 


30 


14 


5 


14 


2 


40 


19 


3 


19 





50 


24 


1 


23 


7 



28 

2.8 

3.2 

3.7 

4.2 

4.6 

9.3 

14.0 

18.6 

23. S 





31 


20 


6 


2.1 


2-01 


7 


2 


4 


2 


4 


8 


2 


8 


2 


7 


9 


3 


1 


3 


1 


10 


3 


5 


3 


4 


20 


7 





6 


8 


30 


10 


5 


10 


2 


40 


14 





13 


6 


50 


17 


5 


17 


1 



20 

2.0 

2.1 

2.6 

30 

3.3 

6.6 

10.0 

13.3 

16.6 





8 , 


6 


0.8| 


7 





g 


8 


1 





9 


1 


2 


10 


1 


3 


20 


2 


6 


30 


4 





40 


5 


3 


50 


6 


6 



0.9 
1.0 
1.1 
1.2 
2.5 
37 
50 
6.2 



P. P. 



12r 



623 



58** 



TABLE VII.- 



32" 



-LOGARITHMIC SINES. COSINES. TANGENTS, 
AND COTANGENTS. 



147* 



' Log, Sin. d, Log. Tan. c. d. Log. Cot. Log. Cos 



72 421 
72 441 
72 461 
72 481 
72 501 



72 522 
72 542 
72 562 
72 582 
72 602 



72 622 
72 642 
72 662 
72 682 
72 702 



72 723 
72 743 
72 763 
72 783 
72 802 



72 822 
72 842 
72 862 
72 882 
72 902 



72 922 
72 942 
72 962 

72 982 

73 002 



73 021 
73 041 
73 06l 
73 081 
73 101 



73 120 
73 140 
73 160 
73 180 
73 199 



73 219 
73 239 
73 258 
73 278 
73 298 



73 317 
73 337 
73 357 
73 376 
73396 



73 415 
73 435 
73 455 
73 474 
73 494 



73 513 
73 533 
73 552 
73 572 
73 591 



9-73 611 
Log. Cos. 



579 
607 
635 
663 
691 

719 
747 
775 
803 
831 
859 
887 
915 
943 
971 
999 
027 
055 
083 
111 
139 
167 
195 
223 
251 
279 
307 
335 
363 
391 

418 
446 
474 
502 
530 
558 
586 
613 
641 
669 

697 
725 
752 
780 
808 
836 
864 
891 
919 
947 
975 
002 
030 
058 
085 
113 
141 
168 
196 
224 
251 
Cot. 



c.d. 



20 421 




20 140 
20 112 
20 084 
20 056 
20 028 



20 000 
19 972 
19 944 
19 916 
19 888 




19 442 
19 414 
19 386 
19 358 
19 330 



19 303 
19 275 
19 247 
19 219 
19 191 



19 164 
19 136 
19 108 
19 080 
19 053 
19 025 
18 997 
18 970 
18 942 
18 914 



18 886 
18 859 
18 83l 
18 803 
18 776 



18 748 



Log. Tan, 



92 842 
92 834 
92 826 
92 818 
92 810 



92 802 
92 794 
92 786 
92 778 
92771 



92 763 
92 755 
92 747 
92 739 
92 731 



92 723 
92 715 
92 707 
92 699 
92 691 



92 643 
92 635 
92 627 
92 619 
92 611 
92 603 
92 595 
92 587 
92 579 
92 570 



92 562 
92 554 
92 546 
92 538 
92 530 



92 522 
92 514 
92 506 
92 498 
92 489 



92 481 
92 473 
92 465 
92 457 
92 449 



92 441 
92 433 
92 424 
92 416 
92 408 



92 400 
92 392 
92 383 
92 375 
92 367 



IS^** 



9-92 359 
Log. Sin. 

624 



P.P. 





28 


38 


37 


6 


2.8 


2.8 


2. 


7 


3 


3 


3.2 


3. 


8 


3 


8 


3-7 


3. 


9 


4 


3 


4-2 


4. 


10 


4 


7 


4.6 


4. 


20 


9 


5 


9.3 


9. 


30 


14 


2 


14.0 


13. 


40 


19 





18.6 


18. 


50 


23 


7 


23.3 


22. 





^<l 


30 


19 


6 


2.0 


2.0 


1.9 


7 


2 


4 


2.3 


2.3 


8 


2 


7 


2.6 


2-6 


9 


3 


1 


3.0 


2.9 


10 


3 


4 


3.3 


3.2 


20 


6 


3 


6.6 


6.5 


30 


10 


2 


10.0 


9.7 


40 


13 


6 


13.3 


13.0 


50 


17 


1 


16.6 


16.2 





8 


8 


6 


0-8 


0-8 


7 


1 





0.9 


8 


1 


1 




9 


1 


3 


1 • 2 


10 


1 


4 


1.3 


20 


2 


8 


2.6 


30 


4 


2 


4.0 


40 


5 


Q 


5.3 


50 


7 


1 


6.6 



0.9 
1.0 
1.1 
1.2 
2.5 
3.7 
5.0 
6.2 



P.P. 



33** 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS. 
AND COTANGENTS. 



146° 



Log. Sin. 



73 611 
73 630 
73 650 
73 669 
73 68 8 



73 708 
73 727 
73 746 
73 766 
73 785 



73 805 
73 824 
73 843 
73 862 
73 882 
73 901 
73 920 
73 940 
73 959 
73 978 



73 997 

74 016 
74 036 
74 055 
7JL074 
74 093 
74 112 
74 131 
74 151 
74 170 



74 189 
74 208 
74 227 
74 246 
74 265 



74 284 
74 303 
74 322 
74 341 
74 360 



74 379 
74 398 
74 417 
74 436 
74 455 



74 474 
74 493 
74 511 
74 535 
74 549 




9-74 756 



Log. Cos 



Log. Tan. c. d. Log. Cot. Log. Cos 



82 
Log. 



251 
279 
307 
334 
362 
390 
417 
445 
473 
500 
528 
555 
583 
610 
638 
666 
693 
721 
748 
776 
803 
831 
858 
886 
913 

941 
968 
996 
023 
051 
078 
105 
133 
160 
188 
215 
243 
270 
297 
325 
352 
380 
407 
434 
462 
489 
516 
544 
571 
598 
626 
653 
680 
708 
735 
762 
789 
817 
844 
871 
898 
Cot. 



18 748 
18 720 
18 693 
18 665 
18 637 



18 610 
18 582 
18 555 
18 527 
18 499 



18 472 
18 444 
18 417 
18 389 
18362 



18 334 
18 306 
18 279 
18 251 
18224 



18 196 
18 169 
18 141 
18 114 
18 086 



18 059 
18 031 
18 004 
17 976 
17 949 



17 921 
17 894 
17 867 
17 839 
17 812 



17 784 
17 757 
17 729 
17 702 
17 675 



17 647 
17 620 
17 593 
17 565 
17 538 



17 510 
17 483 
17 456 
17 428 
17 401 



17 374 
17 347 
17 319 
17 292 
17 265 



17 237 
17 210 
17 183 
17 156 
17 128 



17 101 
Log. Tan, 



92 359 
92 351 
92 342 
92 334 
92 3 26 
92 318 
92 310 
92 301 
92 293 
92 285 



92 277 
92 268 
92 260 
92 252 
92 244 



92 235 
92 227 
92 219 
92 210 
92 202 



92 194 
92 185 
92 177 
92 169 
92 160 



92 152 
92 144 
92 135 
92 127 
92 119 
92 110 
92 102 
92 094 
92 085 
92 077 



92 069 
92 060 
92 052 
92 043 
92 035 



92 027 
92 018 
92 010 
92 001 
91 993 



91 984 
91 976 
91 967 
91 959 
91 951 



91 942 
91 934 
91 925 
91 917 
91 908 



91 900 
91 891 
91 883 
91 874 
91 866 



9-91 857 
Log. Sin 



P. P. 





28 


37 


27 


6 


2.8 


2.7 


2.7 


7 


3 


2 


3 


2 


3 


1 


8 


3 


7 


3 


6 


3 


6 


9 


4 


2 


4 


1 


4 





10 


4 


6 


4 


6 


4 


5 


20 


9 


3 


9 


1 


9 





30 


14 





13 




13 


5 


40 


18 


6 


18 


3 


18 





50 


23 


3 


22 


9 


22 


5 





19 


19 


18 


6 


1-9 


1.9 


1. 


7 


2 


3 


2 


2 


2. 


8 


2 


6 


2 


5 


2. 


9 


2 


9 


2 


8 


2. 


10 


3 


2 


3 


1 


3. 


20 


6 


5 


6 


3 


6. 


30 


9 


7 


9 


5 


9- 


40 


13 




12 


6 


12. 


50 


16 


2 


15 


8 


15. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



8_ 8 



0.8 





1.0 





1.1 


1 


1.3 


1 


1 -4 


1 


2.8 


2. 


4.2 


4 


5.6 


5 


7.1 


6 



P.P. 



123^ 



625 



56* 



TABLE VII. 



34'' 



-LOGARITHMIC SINES. COSINES, TANGENTS. 
AND COTANGENTS. 



145'» 




P. P. 





27 


27 


6 


2-7 


2-7 


7 


3 


2 


3.1 


8 


3 


6 


3-6 


9 


4 


1 


4.0 


10 


4 


6 


4.5 


20 


9 


1 


9.0 


30 


13 


7 


13.5 


40 


18 


3 


18.0 


50 


22 


9 


22.5 



10 
20 
30 
40 
50 



26 

2.6 

31 

35 

4.0 

4.4 

8.8 

13.5 

17.6 

22.1 



19 18_ 18 

8 
• 1 
4 
7 



.0 
.0 



1 


9 


1 


8 


1. 


2 


2 


2 


1 


2. 


2 


5 


2 


4 


•2 


2 


8 


2 


8 


2 


3 


1 


3 


1 


3 


6 


3 


6 


1 


6 


9 


5 


9 


2 


9. 


12 


5 


12 


3 


12 


15 


8 


15 


4 


15. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



9 8_ 
8 

1 
3 
4 






9 





1 





1 


1 


2 


1 


1 


3 


1 


1 


5 


1 


3 





2 


4 


5 


4 


6 





5 


7 


5 


7 



P. p. 



1«4* 



620 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 



144° 



' Log' Sin, d. Log. Tan.[c.d.[Log. Cot. Log. Cos. d 



• 75 859 

• 75 877 

• 75 895 

• 75 913 
.75 931 



.75 949 
.75 967 

• 75 985 

• 76 003 

• 76 021 



•76 039 
•76 057 
•76 075 
•76 092 
76 110 



•76 128 
•76 146 
•76 164 
•76 182 
•76 200 



•76 217 
•76 235 
•76 253 
•76 271 
•76 289 



•76 306 
•76 324 
•76 342 
•76 360 
•76 377 
•76 395 
•76 413 
•76 431 
•76 448 
76 466 



9-76 484 
9-76 501 
9-76 519 
9-76 536 
9 • 76 554 



84 925 
9-84 952 
9-84 979 
9.85 005 
9-85 032 



0-15 477 
0-15 450 
0-15 423 
0-15 396 
0.15 370 



9. 76 572 
9.76 589 
76 607 
9-76 624 
9-76 642 



9-76 660 
9-76 677 
9-76 695 
9-76 712 
9-76 730 



9-76 747 
9-76 765 
9-76 782 
9-76 800 
9-76 817 
9.76 835 
9.76 852 
9-7,6 869 
9.76 887 
9. 76 904 



9-85 059 
9. 85 086 
9-85 113 
9-85 139 
9 ^85 166 
9^85 193 
9-85 220 
9-85 246 
9-85 273 
9-85 300 



9-85 327 
85 353 
9-85 380 
9-85 407 
9-85 4 33 
9-85 460 
9-85 487 
9-85 513 
9-85 540 
9-85 567 



0-15 343 
0-15 316 
0-15 289 
0-15 262 
0-15 235 



-91 336 
•91 327 
•91 318 
•91 310 
•91 301 



-91 292 
-91 283 
-91 274 
91 265 
•91 256 



•91 247 
• 91 239 
-91 230 
-91 221 
-91 212 



•91 203 
•91 194 
•91 185 
•91 176 
•91 167 



0-14 807 
0-14 780 
0-14 753 
0-14 726 
0-14 700 



9-85 594 
9-85 620 
9-85 647 
9-85 673 
9-85 700 



9-76 922 



Log. Cos. 



9-85 727 
9-85 753 
9-85 780 
9-85 807 
9. 85 833 



9.85 860 
9-85 887 
9.85 913 
9.85 940 
9.85 966 



9.85 993 

9.86 020 
9.86 046 
- 86 073 
9^8X099 
9-86 126 



0-14 940 
0-14 914 
0-14 887 
0-14 860 
0-14 833 



91 158 
•91 149 
•91 140 
•91 131 
•91 122 



0-14 673 
0-14 646 
0-14 620 
0-14 593 
0. 14 566 



•14 539 
•14 513 
•14 486 
•14 459 
•14 433 



0-14 406 
0-14379 
0.14353 
0.14326 
0JJL299 

0Tl4~273 
0.14 246 
0.14219 
0.14 193 
0.14 166 



d. Log. Cot. c d 



•91 113 
•91 104 
•91 095 
•91 086 
■91 077 



• 91 068 
•91 059 
•91 050 
•91041 
•91 032 



•91 023 
•91 014 
•91 005 

• 90 996 

• 90 987 



0^14 140 
0-14 113 
0.14086 
014 060 
0. 14033 



0.14 007 
0-13 980 
0.13 953 
0-13 927 
13 900 



013 874 



Log. Tan, 



• 90 978 
•90 969 
•90 960 
•90 951 

• 9 942 
9.90 933 
9.90 923 
9-90 914 
9-90 905 
9.90 896 



390 887 
9.90 878 
9. 90 869 
9.90 860 
9-90 850 



9.90 841 
9.90 832 
9.90 823 
9.90 814 
9_90 805 
9-90 796 



Log. Sin.) d. 
~627 



P. P. 





27 


26 


6 


2.7 


2.6 


7 


3 


1 


3 


1 


8 


3 


6 


3 


5 


9 


4 





4 





10 


4 


5 


4 


4 


20 


9 





8 


8 


30 


13 


5 


13 


2 


40 


18 





17 


6 


50 


22 


5 


22 


1 





18 


17 


17 


6 


1.8 


1-7 


1- 


7 


2.1 


2 





2. 


8 


2-4 


2 


3 


2. 


9 


2-7 


2 


6 


2. 


10 


3-0 


2 


9 


2. 


20 


6-0 


5 


8 


5. 


30 


9-0 


8 


7 


8. 


40 


12-0 


11 


6 


U- 


50 


15-0 


14 


6 


14- 





9 


9 


S 


6 


0-9 


0-9 


08 


7 


1-1 


1-0 


10 


8 


1-2 


1-2 


1.1 


9 


1-4 


1-3 


1^3 


10 


1-6 


1-5 


1^4 


20 


3-1 


3-0 


2-8 


30 


4-7 


4..'^ 


4-2 


40 


6-3 


6-0 


5-6 


50 


7.9 


7.5 


7.1 



P.P. 



54' 



TABLE VII.- 



36'' 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



143* 



Log. Sin, d. Log. Tan. c.d. Log. Cot. Log. Cos, d. 



• 76 922 

• 76 939 

• 76 956 

• 76 974 

• 76 991 



• 77 095 

• 77 112 

• 77 130 

• 77 147 

• 77 164 



•77 181 
•77 198 
•77 216 
•77 233 
• 77 250 



•77 267 
• 77 284 
•77 302 
•77 319 
77 336 



•77 353 
•77 370 

• 77 387 

• 77 404 

• 77 421 



• 77 439 
•77 456 
•77 473 
•77 490 
•77 507 
•77 524 
■77 541 
•77 558 
•77 575 
77 592 



• 77 609 
•77 626 

• 77 643 
•77 660 
•77 677 



■ 77 693 
77 710 

• 77 727 

• 77 744 

■ 77 76l 



■ 77 778 
• 77 795 
■77 812 
■77 828 
77 845 



■ 77 862 

■ 77 879 

■ 77 896 
.^77 913 

■ 77 929 



9 77 946 



Log. Cos 



9 86 126 
86 152 
9-86 179 
9 86 206 
9 86 232 



9^86 259 
986 285 
986 312 
9^86 338 
986 365 



9-86 391 
9-86 418 
9-86 444 
9-86 471 
9-86 49 7 
9-86 524 
9-86 550 
986 577 
86 603 
86 630 



9^86 656 
9^86 683 
86 709 
9.86 736 
9-86 762 



9 86 788 
9.86 815 
9.86 841 
9-86 868 
986 894 



9-86 921 
9-86 947 
9^86 973 
9 87 000 
9^87 026 
9-87 053 
9^87 079 
9-87 105 
9^87 132 
9-87 158 



9^87 185 
9^87 211 
9-87 237 
9^87 264 
9-87 290 



87 316 

87 343 

9^87 369 

9^87 395 

9-87 422 



87 448 
87 474 
9-87 501 
9-87 527 
9^87 55 3 
9 87 580 
9 87 606 
9. 87 632 
987 659 
9. 87 685 



9 87 71] 



013 874 
0^13 847 
0-13 821 
0.13 794 
0-13 767 



• 13 741 
.13 714 
•13 688 
•13 661 
•13 635 



13 608 

13 582 

0^13 555 

0^13 529 

13 502 



5 9 



• 13 476 

• 13 449 
.13 423 
•13 396 

• 13 370 



0.12 815 
0.12 789 
0.12 762 
0.12 736 
0.12 709 



0.12 683 
0.12 657 
0.12 630 
0.12 604 
0.12 578 



0-12 420 
0.12 393 
012 367 
0^12 341 
0-12 315 
0^12 288 



• 90 657 

• 90 648 

• 90 639 

• 90 629 

• 90 620 



• 90 611 

• 90 602 
.90 592 
.90 583 

• 90 574 



• 90 564 

• 90 555 

• 90 546 

• 90 536 
■ 90 527 

• 90 518 

• 90 508 

• 90 499 

• 90 490 

• 90 480 



• 90 471 

• 90 461 

• 90 452 

■ 90 443 

■ 90 433 



5 9 



• 90 377 

• 90 367 

• 90 358 
.90 348 

90 339 



• 90 330 

• 90 320 

• 90 311 

• 90 30l 

• 90 292 



i^e** 



Log. Cot.|c.d. Log. Tan, Log. Sin 

628 



P. P. 



10 
20 
30 
40 
50 



27 


26 


2.7 


2.6 


3.1 


3.1 


3.6 


3.6 


4-0 


4-0 


4.5 


4.4 


9.0 


8-8 


13.5 


13.2 


18.0 


17.6 


22.5 


22.1 





17 


17 


6 


1.7 


1.7 


7 


2 





2.0 


8 


2 


3 


2-2 


9 


2 


6 


2.5 


10 


2 


9 


2^8 


20 


5 


8 


5.6 


30 


8 


7 


8.5 


40 


11 


6 


11.3 
14.1 


50 


14 


6 



16_ 

1.6 
1.9 

2.2 
2.5 
2.7 
5.5 
8.2 
11.0 
13.^ 





9 


9 


6 


0.9 


0.9 


7 


1.1 




8 


1.2 


1 .2 


9 


1-4 


1 "3 


10 


1^6 


1.5 


20 


31 


3.0 


30 


4.7 


4.5 


40 


6.3 


6.0 


50 


7.9 


7.5 



P.P. 



63" 



37** 



TABLE VIT.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



14.^ 



' 


Log. Sin. 


d. 


Log. Tan 


c.d. 


Log. Cot. 


Log. Cos, 


d. 




P. P. 





9-77 946 


16 
17 
16 
17 


9^87 711 


26 
26 
26 
26 


]0.12 288 


9.90 235 


9 
10 


60 




1 


9-77 963 


9-87 737 


0.12 262 


9.90 225 


59 




2 


9-77 980 


9^87 764 


0.12 236 


9.90 216 


58 




3 

4 


9-77 996 
9.78 013 


9^87 790 
9^87 816 


0.12 209 
0.12 183 


9.90 206 
9.90 196 


57 
56 




5 


9. 78 030 


16 
16 
17 
16 
17 


9 87 843 


2b 
26 
26 
26 
26 


0.12 157 


9.90 187 


9 
9 
9 


55 




6 


3-78 046 


9-87 869 


;0.12 131 


9.90 177 


54 




7 


3.78 063 


9. 87 895 


0.12 104 


9 90 168 


53 




8 < 


3.78 080 


9-87 921 


0.12 078 


9.90 158 


52 


26 ^ft 


9 < 


3.78 097 


9 87 948 


0.12 052 


9 90 149 


51 


6 5t.RI 


26 


10 ( 


3.78 113 


lb 
16 

11 

16 

16 
16 
17 
16 
16 

16 
16 
16 
16 
17 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

16 
16 
16 
16 
16 


9. 87 974 


2b 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 


0-12 026 


9^90 139 


9 
9 

10 
9 
9 
9 
9 

10 
9 
9 
9 

10 
9 
9 

10 

1 
9 
10 
9 
9 

10 

9 

10 

9 

10 

10 

9 
10 

9 
10 

9 
10 

9 
10 

9 
10 
10 

10 
9 

10 

10 
9 

10 

10 

10 
10 


50 


7 


3 


•1 


3.0 


11 1 


3-78 130 


9-88 000 


0.11 999 


9 90 130 


49 


8 


3 


■ 5 


3.4 


12 i 


3.78 147 


9^88 026 


0.11 973 


9^90 120 


48 


9 


4 


• 


3.9 


13 i 


).78 163 


9 88 053 


0.11 947 


9 90 110 


47 


10 


4 


•4 


4-3 


14 i 


).78 180 


988 079 


0.11 921 


9. 90 101 


46 
45 


20 
30 


8 

13 


:ii 


8.6 


]5 S 


).78 196 


9-88 105 


0.11 895 


9-90 091 


30 


16 S 


).78 213 
).78 230 


9-88 131 


0.11 868 


9-90 082 


44 


40 


IV 


•^li-'i 


17 S 


9 88 157 


0.11 842 


9.90 072 


43 


50 iiz-il^ii.o 


18 t 


).78 246 


9 88 184 


0.11 816 


9. 90 062 


42 




Hi 


•78 263 


9. 88 210 


0-11 790 


9.90 053 


41 




30 £ 


•78 279 


9. 88 236 


10-11 763 


9^90 043 


40 




21 £ 


1-78 296 


9 88 262 


0-11 737 


9^90 033 


39 




22 i 


178 312 


9^88 288 


0-11 711 


9^90 024 


38 




23 i 


178 329 


988 315 


0-11 685 


9^90 014 


37 




24 i 


).78 346 


9. 88 341 


Oil 659 


9 . 90 004 


11 
35 




25 S 


).78 362 


9. 88 367 


0-11 633 


9^89 995 




26 i 

27 £ 


1.78 379 
178 395 


9-88 393 
9-88 419 


0-11 606 
0.11 580 


9 89 985 
9 89 975 


34 
33 


6 


17 

17 


1( 

1- 


i 16 

6 1.6 


28 i 


1.78 412 


9 • 88 445 


0.11 554 


9. 89 966 


32 


7 


2 


• 


1- 


9 1.8 


29 t 


1-78 428 


9. 88 472 


0-11 528 


9. 89 956 


31 
80 


8 

9 
10 
20 
30 
40 
50 


2 
2 
2 
5 
8 


• 5 
•8 


2. 
2. 
2. 
5. 
8^ 


2 2.1 


30 £ 


- 78 444 


9-88 498 


0^11 502 


9^89 946 


5 2.4 
7 2.6 
5 5.3 
2 8.0 


31 t 


-78 461 


9^88 524 


0.11 476 


9^89 937 


29 


32 S 


•78 477 


9.88 550 


0-11 449 


9^89 927 


28 


• 6 
•5 


33 J 


-78 494 


9^88 576 


0.11 423 


9^89 917 


27 


34 8 


78 510 


9-88 602 


0-11 397 


9. 89 908 


26 
25 


11 

1 A 


•3 

T 


11^ 

1 o 


10.6 
7 13.3 


35 9 


-78 527 


9.88 629 


0^11 371 


9^89 898 




36 y 


•78 543 


9.88 655 


0.11 345 


9.89 888 


24 




37 y 


-78 559 


9. 88 681 


0.11 319 


9. 89 878 


23 




38 y 


•78 576 


9^88 707 


0.11 293 


9. 89 869 


22 




39 9 


•78 592 


9-88 733 


0.11 266 


9.89 859 


21 
20 




40 9 


•78 609 


9^88 759 


0^11 240 


9^89 849 




41 y 


-78 625 


9. 88 785 


0^11 214 


9^89 839 


19 




42 9 


-78 641 


9. 88 8ll 


0.11 188 


9.89 830 


18 




43 y 


-78 658 


9. 88 838 


0.11 162 


9.89 820 


17 




44 9 


•78 674 


9. 88 864 


0.11 136 


9-89 810 
9.89 800 


16 
15 


10 


% 


45 9 


• 78 690 


9. 88 890 


0.11110 


6 1 " " 

71 

81 

91 
101. 
20 3- 
30 5^ 


U U 
I 1 
3 1 
5 1 

e'l 

33 
4 


9 

■\ 

4 

:! 

• 7 


46 9 


• 78 707 


9. 88 916 


0.11 084 


9^89 791 


14 


47 9 


•78 723 


9 •88 942 


0.11 058 


9. 89 781 


13 


48 9 


•78 739 


9. 88 968 


0-11 032 


9-89 771 


12 


49 9 


• 78 755 


9. 88 994 


0^11 005 


9-89 761 


10 


60 9 


-78 772 


9-89 020 


0^10 979 


9^89 75l 


51 9 


• 78 788 


9-89 046 


)^10 953 


9^89 742 


9 


40 6^ 


6 B 


.3 


52 9 


• 78 804 


9^89 072 


0^10 927 


9^89 732 


8 


50 8 • 


3 7 


.9 


53 9 


-78 821 


9. 89 098 


010 901 


9.89 722 


7 




54 9 


• 78 837 


9. 89 124 


0^10 875 


9. 89 712 


6 
5 




55 9 


• 78 853 


9. 89 150 


0-10 849 


9.89 702 




56 c 


■ 78 869 


9-89 177 


3^10 823 


9^89 692 


4 




57 9 


• 78 885 


9^89 203 


0^10 797 


9-89 683 


3 




58 c 


• 78 902 


9^89 229 


0.10 771 


989 673 


2 




59 9 


• 78 918 


9.89 255 


10 745 


9^89 663 


1 




60 9 


• 78 934 


16 


9. 89 281 '"^^ 


0.10 719 


9 89 653 


lU 







■ L 


og. Cos. 


d. 


Log. Cot.ic d. 


Log. Tan. 


Log. Sin. 


d. 


/ 


P.P. 



137* 



629 



SS** 



as** 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



141° 



Log, Sin. d, 



78 934 
78 950 
78 966 
78 982 
78 999 



79 015 
79 031 
79 047 
79 063 
79 079 



79 095 
79 111 
79 127 
79 143 
79 159 



79 175 
79 191 
79 207 
79 223 
79 239 




79 494 
79 510 
79 526 
79 541 
79 557 



79 573 
79 589 
79 605 
79 620 
79 636 



79 652 
79 668 
79 683 
79 699 
79 715 



79 730 
79 746 
79 762 
79 777 
79 793 



79 809 
79 824 
79 845 
79 856 
79 871 



9 79 887 



Log. Cos. 



Log, Tan. c.d. Log. Cot. Log, Cos, di 



89 281 
■ 89 307 
89 333 
89 359 
89 385 



89 411 
89 437 
89 463 
89 489 
89 515 



89 541 
89 567 
89 593 

• 89 619 

• 89 645 



• 89 671 
89 697 
89 723 
89 749 

•89 775 



89 801 
89 827 
89 853 
89 879 
89 905 



89 931 
89 957 

89 982 

90 008 

■ 90 034 
90 060 
90 086 

■ 90 112 
90 138 

■ 90 164 



90 190 

■ 90 216 

■ 90 242 
90 268 

• 90 294 



■ 90 319 

■ 90 345 

■ 90 371 
90 397 

• 90 423 



• 90 449 
90 475 
90 501 

• 90 526 

• 90 552 



• 90 578 
90 604 

• 90 630 

• 90 656 

• 90 682 



• 90 707 
90 733 

• 90 759 

• 90 785 

• 90 811 



• 90 837 

Log. Cot. c.d 



■ 10 589 

• 10 563 
10 537 

■ 10 511 

• 10 485 



• 10 459 
10 433 

■ 10 407 

• 10 381 

• 10 355 



• 10 329 

■ 10 303 

■ 10 277 
10 251 

• 10 225 



■ 10 199 

• 10 173 

• 10 147 

• 10 121 

• 1 095 

• 10 069 

• 10 043 

■ 10 017 

■ 09 991 

■ 09 965 



09 939 
• 09 913 

■ 09 887 

■ 09 86l 
^9_836 

■ 09 810 
09 784 

■ 09 758 
■09 732 
■09 706 



■ 09 680 

■ 09 654 

■ 09 628 

■ 09 602 
09 577 



■ 09 551 

• 09 525 

• 09 499 
09 473 

• 09 447 



09 421 

■ 09 395 

■ 09 370 

■ 09 344 
• 09 318 



• 09 292 

• 09 266 
■09 240 

09 214 

■0 9 189 

•09 163 

Log. Tan 



89 604 
• 89 594 

89 584 
■ 89 574 
•89 564 



• 89 554 
■ 89 544 

89 534 

• 89 524 
■89 514 



■ 89 504 

■ 89 494 
89 484 
89 474 

■ 89 464 



89 253 
89 243 

■ 89 233 
89 223 

■89 213 



• 89 203 

■ 89 193 
89 182 

■ 89 172 

■ 89 162 



89 152 

■ 89 142 

■ 89 132 
• 89 121 

I 111 



89 101 

• 89 091 
■ 89 081 

• 89 070 

• 89 060 



138° 



9-89 050 
Log. Sin. 

630 



P. P. 



6| 

7 

81 

91 3 
10 4 
20' 8 
3013 
4017 
5021 



26 

2.6 



25_ 

2.5 

3-0 

3.4 

3-8 

4.2 

8.5 

12.7 

17.0 

21.2 





16 


16 


1^ 


6 


1.6 


1^6 


1 


7 


1 


9 


1 


8 


1 


8 


2 


2 


2 


1 


2 


9 


2 


5 


2 


4 


2. 


10 


2 


7 


2 


6 


2. 


20 


5 


5 


5 


3 


5. 


30 


8 


2 


8 





7. 


40 


11 





10 


6 


10. 


50 


13 


7 


13 


3 


12. 



10 10 9 

9 
1 
2 
4 


















2 




1 


1. 




4 




3 


1. 




6 




5 


1. 




7 




6 


1. 


3 


5 


3 


3 


3. 


5 


2 


5 





4. 


7 





6 




6. 


8 


7 


8 


3 


7. 



P. p. 



51« 



39*' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



140^ 



Log. Sin. d. Log. Tan. c.d. Log. Cot. Log. Cos 



79 887 
79 903 
79 918 
79 934 
79 949 



79 965 
79 980 

79 996 

80 011 
80 027 



80 042 
80 058 
80 073 
80 089 
80 104 



80 120 
80 135 
80 151 
80 166 
80 182 



80 197 
80 213 
80 228 
80 243 
80 259 




80 427 
80 443 
80 458 
80 473 
80 488 



80 504 
80 519 
80 534 
80 549 
80 564 



80 580 
80 595 
80 610 
80 625 
80 640 



80 655 
80 671 
80 686 
80 701 
80 716 



80 731 
80 746 
80 761 
80 776 
80 791 



80 806 
Log. Cos, 



90 837 
90 863 
90 888 
90 914 
90 940 



90 966 

90 992 

91 017 
91 043 
91 069 



91 095 
91 121 
91 146 
91 172 
91 198 



91 224 
91 250 
91 275 
91 301 
91 327 



91 353 
91 378 
91 404 
91 430 
91 456 



91481 
91 507 
91 533 
91 559 
91 584 



91 610 
91 636 
91 662 
91 687 
91 713 



91 739 
91 765 
91 790 
91 816 
91 842 



91 867 
91 893 
91 919 
91 945 
91 970 



91 996 

92 022 
92 047 
92 073 
9 2 099 
92 124 
92 150 
92 176 
92 201 
92 227 



92 253 
92 278 
92 304 
92 330 
92 355 



92 381 
Log. Cot. 



-08 776 
08 750 

■ 08 724 
08 698 

■ 08 673 



08 389 
• 08 364 
08 338 

■ 08 312 

■ 08 286 



• 08 261 
08 235 
08 209 

■ 08 183 

• 08 158 



■ 08 132 

• 08 106 

■ 08 081 

• 08 055 

■ 08 02^ 



• 08 004 

• 07 978 
■ 07 952 

07 926 

• 07 901 



• 07 875 

• 07 849 

• 07 824 

• 07 798 

• 07 772 



• 07 747 

• 07 721 

■ 07 695 

■ 07 670 
lOZ144 

• 07 618 
Log. Tan. 



89 050 

■ 89 040 
89 030 
89 019 

■ 89 009 



• 88 999 

■ 88 989 

■ 88 978 

■ 88 968 

• 88 958 



• 88 947 
88 937 

• 88 927 
88 917 
88 906 



88 896 
88 886 
88 875 
■ 88 865 
88 855 



• 88 792 

• 88 782 
88 772 

.88 761 

• 88 751 



• 88 636 
■ 88 625 

88 615 

• 88 604 
88 594 



■ 88 583 

■ 88 573 
88 562 

• 88 552 
88 541 

•88 531 

• 88 520 

• 88 510 

• 88 499 

• 88 489 



• 88 47g 

• 88 467 

• 88 457 

• 88 446 
■ 88 436 



139'' 



88 425 
Log. Sin. 



60 

59 
58 
57 
16 
55 
54 
53 
52 
11 
50 
49 
48 
47 

ii 

45 
44 
43 

42 

il 
40 

39 
38 
37 
11 
35 
34 
33 
32 
11. 
30 
29 
28 
27 
21 
25 
24 
23 
22 
21_ 

20 

19 

18 

17 

11 

15 

14 

13 

12 

_11 

10 

9 

8 

7 

_6 

5 

4 

3 

2 

_1 

O 



P. p. 



26 



7 
8 
9 
10 
20 
30 
40 
50 



2 


6 


2 


3 





3 


3 


4 


3 


3 


9 


3 


4 


3 


4 


8 


6 


8 


13 





12 


17 


3 


17 


21 


6 


21 



25 

5 

4 
8 
2 
5 
7 

2 





1 


B 


15 


1 


6 


1^6 


1-5 


1- 


7 


1 


8 


1 


8 


1. 


8 


2 


1 


2 





2. 


9 


2 


4 


2 


3 


2 


10 


2 


6 


2 


6 


2 


20 


5 


3 


5 


1 


5 


30 


8 





7 


7 


7 


40 


10 


6 


10 


3 


10 


50 


13 


3 


12 


9 


12 



11 

1.1 



10 10 

i-ojio 

.21. 
.4!l. 
.611. 
.7il. 
.53. 
• 215. 
06. 
.7'8. 



P.P. 



60° 



TABLE VII. 



40° 



-LOGARITHMIC SINEb3, COSINES, TANGENTS, 
AND COTANGENTS. 



139° 



Log. Sin. I d. 



80 806 
80 822 
80 837 
80 852 
80 867 



80 882 
80 897 
• 80 912 
80 927 
80 942 



10 

11 
12 
13 
14 
15 
16 
17 
18 

11 
20 

21 
22 
23 
24. 
25 
26 
27 
28 
29_ 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 
45_ 
46 
47 
48 
4i 
50 
51 
52 
53 
5£ 
55 
56 
57 
58 
59. 
60 



).80 957 
).80 972 
) 80 987 
).81 001 
).81 016 



• 81 031 
81 046 
81 061 
81 076 
81 091 



81 106 

• 81 121 

• 81 136 
81 150 
81 16 5 

•81 180 
81 195 
81 210 

•81 225 

• 81 239 



■81 254 
81 269| 
81 284 
81 299 
81 313 



81 328 
81 343 
81 358 
81 372 
81 387 



81 402 
81 416 
81 431 
81 446 
81 460 



• 81 475 
•81 490 
•81 504 
•81 519 
•81 534 



81 548 
81 563 
81 578 
81 592 
81 607 



81 62l 
81 636 
81 650 
81 665 
81J880 
81 694 



Log. Tan. c. d. Log. Cot. Log. Cos.| d. 



92 381 25 
92 407 -^ 
92 432 
92 458 
92 484 



92 509 
92 535 
92 561 
92 586 
92 612 



92 638 
92 663 
92 689 
92 714 
92 740 



92 766' 
92 791 
92 817 
92 842 
92 838i 



92 894 
92 919 
92 945 
92 971 
92 996 



93 022 
93 047 
93 073 
93 098 
93 124 



93 150 
93 175 
93 201 
93 226 
93 252 



93 278 
93 303 
93 329 
93 354 
93 380 



93 405 
93 431 
93 456 
93 482 
93 508 



93 533 
93 559 
93 584 
93 610 
93 635 



93 661 
93 686 
93 712 
93 737 
93 763 



93 788 
93 814 
93 840 
93 865 
93 891 



93 916 



0-07 618 
007 593 
07 567 
07 541 
007 516 



007 234 
07 208 
007 183 
007 157 
007 13l 



007 106 
0-07 080 
007 055 
0-07 029 
007 003 



07 490 
07 465 
007 439 
007 413 
007 388 
07 362 
007 336 
007 311 
007 285 
007 259 



006 978 
0-06 952 
006 927 
06 901 
006 875 



06 850 
06 824 
06 799 
06 773 
06 748 



006 
006 
006 
06 
006 



722 
696 
671 
645 
620 



006 
006 
006 
006 
006 



594 
569 
543 
518 
492 



006 466 
06 441 
006 415 
006 390 
006 364 



08 339 
006 313 
006 288 
006 262 
006 237 



006 211 
0.06 186 
006 160 
006 134 
006 109 



0-06 083 




88 425 
88 415 
88 404 
88 393 
88 383 



88 372 
88 361 
88 351 
88 340 
88 329 



88 212 
88 201 
88 190 
88 180 
88 1 



88 158 
88 147 
88 137 
88 126 
88 115 



88 104 
88 094 
88 083 
88 072 
88 061 



88 050 
88 039 
88 029 
88 018 
88 007 



87 996 
87 985 
87 974 
87 963 
87 953 
87 942 
87 931 
87 920 
87 909 
87 898 



87 887 
87 876 
87 865 
87 854 
87 844 



87 833 
87 822 
87 811 
87 800 
87 789 
87 778 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 

ii 

45 
44 
43 

42 
41 



P.P. 



36 

2 



10 4 
20 8 
30 13 
40 17 
50 21 



25 

2-5 





15 


15 


1? 


6 


1.5 


1-5 


1.4 


7 


1 


8 


1 


7 


1 


7 


8 


2 





2 





1 


9 


9 


2 


3 


2 


2 


2 


2 


10 


2 


6 


2 


5 


2 




20 


5 


1 


5 





4 


8 


30 


7 


7 


7 


5 


7 


2 


40 


10 


3 


10 





9 


g 


50 


12 


9 


12 


5 


12 


1 





11 


10 


6 


1.1 


i.b 


7 


1 


3 


1.2 


8 


1 


4 


1.4 


9 


1 


6 


1.6 


10 


1 


8 


1.7 


20 


3 


6 


35 


30 


5 


5 


5.2 


40 


7 


3 


7.0 


50 


9 


1 


8.7 



Log. Cos. 



Log. Cot. c.d. Log. Tan. 



-og. Sin. 



P. P. 



130° 



(3.32 



49^ 



41^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



' Log. Sin, 




81 694 
81 709 
81 723 
81 738 
81 752 



81 767 
81 781 
81 796 
81 810 
81824 



81 983 

81 997 

82 012 
82 026 
82 040 



82 055 
82 069 
82 083 
82 098 
82 112 



82 126 
82 140 
82 155 
82 169 
82 183 



82 197 
82 212 
82 226 
82 240 

82 254 



82 269 
82 283 
82 297 
82 311 
82 325 



82 339 
82 354 
82 368 
82 382 
82 396 



82 410 
82 424 
82 438 
82 452 
82 467 



82 481 
82 495 
82 509 
82 523 
82 537 
82 551 
Log. Cos. 



Log. Tan. c. d 



93 916 
93 942 
93 967 

93 993 

94 018 
94 044 
94 069 
94 095 
94 120 
94 146 



94 171 
94 197 
94 222 
94 248 
94 273 



94 299 
94 324 
94 350 
94 375 
94 400 



94 426 
94 451 
94 477 
94 502 
94 528 



94 553 
94 579 
94 604 
94 630 
94 655 



94 681 
94 706 
94 732 
94 757 
94 782 



94 808 
94 833 
94 859 
94 884 
94 910 



94 935 
94 961 

94 986 

95 Oil 
95 037 



95 062 
95 088 
95 113 
95 139 
95 164 



95 189 
95 215 
95 240 
95 266 
95 291 



95 316 
95 342 
95 367 
95 393 
95 418 



9-95 443 



Log. Cot, 



Log. Cot. Log. Cos. 



0- 06 083 
.06 058 

• 06 032 

• 06 007 

• 05 981 



c.d, 



05 956 
05 930 
05 905 
05 879 
05 854 



05 828 
05 803 
05 777 
05 752 
05 726 



05 701 
05 675 
05 650 

05 625 
05 P99 



05 574 
05 548 
05 523 
05 497 
05 472 



05 446 
05 421 
05 395 
05 37C 
05 344 



05 319 
05 293 
05 268 
05 243 
05 217 



05 192 
05 166 
05 141 
05 115 
05 090 



05 064 
05 039 
05 014 
04 988 
04 963 



04 937 
04 912 
04 886 
04 861 
04 836 



04 810 
04 785 
04 759 
04 734 
04^8 
04 683 
04 658 
04 632 
04 607 
04 581 



04 556 



Log. Tan, 



87 778 
87 767 
87 756 
87 745 
87 734 



87 723 
87 712 
87 701 
87 690 
87 679 



87 668 
87 657 
87 645 
87 634 
87 623 



87 612 
87 601 
87 590 
87 579 
87 568 



87 557 
87 546 
87 535 
87 523 
87 512 



87 501 
87 490 
87 479 
87 468 
87 457 



87 445 
87 434 
87 423 
87 412 
87 401 



87 389 
87 378 
87 367 
87 356 
87 345 



87 333 
87 322 
87 311 
87 300 
87 288 



87 277 
87 266 
87 254 
87 243 
87 232 



87 221 
87 209 
87 198 
87 187 
87 175 



87 164 
87 153 
87 14l 
87 130 
87 118 



9 87 107 



131° 



Log. Sin. 
633 



11 



138' 



P.P. 





25 


25 


6 


2-5 


25 


7 


3 





2 


9 


8 


3 


4 


3 


3 


9 


3 


8 


3 


7 


10 


4 


2 


4 


1 


20 


8 


5 


8 


3 


30 


12 


7 


12 


5 


40 


17 





16 


6 


50 


21 


2 


20 


8 





1? 


14 


6 


1.4 


1.4 


7 


1 


7 


1-6 


8 


1 


9 


18 


9 


2 


2 


2.] 


10 


2 


4 


2.3 


20 


4 


8 


4-6 


30 


7 


2 


70 


40 


9 


6 


9-3 


50 


12 


1 


11.6 





1 


1 11 




3 




5 




7 




9 




8 




7 




g 


9 


6 



11 

11 

1-3 
l.i 
1-6 
1-8 
3-6 
5-5 
7-3 
9.1 



P.P. 



48^ 



43° 



TABLE VII,— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



137" 



Log. Sin. 



O 

1 
2 
3 

5 
6 
7 
8 

10! 

11 
12 
13 
U 
15 
16 
17 
18 
ii 
20 
21 
22 
23 
2± 

25 
26 
27 
28 
29. 
30 
31 
32 
33 
3± 

35 
36 
37 
38 
39 



551 
565 
579 
593 
607 
621 
635 
649 
663 
677 
691 
705 
719 



733 {1 

760j If 
774' 1^ 



788' 
802 
816 
830 
844 
858 
871 
881' 
8991 
913 
927i 
940 
954i 



40 

41 
42 
43 
44 
45 
46 
47 
48 
^1. 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



982 
996 
009 
023 
037 
051 
064 
07§ 
092 
106 
119 
133 
147 
160 
174 
188 
201 
215 
229 
242 
256 
269 
283 
297 
310 
324 
337 
351 
365 
378 



Log. Cos. d. 



Log. Tan. c, d. Log. Cot. Log. Cos 



95 443 
95 469 
95 494 
95 520 
95 545 



95 571 
95 596 
95 621 
95 647 
95 672 



95 697 
95 723 
95 748 
95 774 
95 799 
95 824 
95 850 
95 875 
95 901 
95 926 



95 951 

95 977 

96 002 
96 027 
96 053 
96 078 
96 104 
96 129 
96 154 
96 18J 
96 205 
96 230 
96 256 
96 281 

96 332 
96 357 
96 383 
96 408 
96 433 



96 459 
96 484 
96 509 
96 535 
96 560 



96 585 
96 611 
96 636 
96 66l 
96 687 




Log. Cot, 



556 
531 
505 
480 
454 
429 
404 
378 
353 
32.7 
30i 
277 
25l 
226 
200 



175 
150 
124 
099 
074 



048 
023 
997 
972 
947 
921 
896 
871 
845 
820 
795 
769 
744 
718 
693 
668 
642 
617 
592 
566 



541 
516 
490 
465 
440 



414 
389 
364 
338 
313 



287 
262 
237 
211 
186 

161 
135 
110 
085 
059 



03 034 



Log, Tan 



87 107 
87 096 
87 084 
87 073 
87 062 



87 050 
87 039 
87 027 
87 016 
87 004 



86 993 
86 982 
86 970 
86 959 
86 947 



86 936 
86 924 
86 913 
86 901 
86 890 



86 647 
86 635 
86 623 
86 612 
86 600 



86 588 
86 577 
86 565 
86 553 
86 542 



86 530 
86 518 
86 507 
86 495 
86 483 



86 471 
86 460 
86 448 
86 436 
86 424 



133** 



986 412 
Log. Sin 

634 



60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
)0 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 

19 
18 
17 
16^ 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 
3 
2 
_1 
O 



P. P. 



7 

8 3 

9 3 

10 4 
20 8 
30,12 
4017 
50121 



25_ 

2.51 



25 

2 5 



5i 8 
7il2 
16 
2I2O 





1 


1 


13 


6 


1 


4 


1.3 


7 


1 


5 


1 


6 


8 


1 


g 


1 


8 


9 


2 


1 


2 





10 


2 


3 


2 


2 


20 


4 


6 


4 


5 


30 


7 





6 


7 


40 


9 


3 


9 





50 


11 


6 


11 


2 





12 


IT 




6 


1.2 


1 1 




7 


1 


4 




3 




8 


1 


6 




5 




9 


1 


8 




7 




10 


2 







9 




20 


4 





3 


8 


3 


30 


6 





5 


7 


5 


40 


8 





7 


6 


7 


50 


10 





9 


6 


9 



P. p. 



47^ 



TABLE VII. 



43° 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



ISC'* 



Log. Sin. 



O 

1 

2 

3 

_4 

5 

6 

7 

8 

J_ 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 




20 

21 

22 

23 

24 

25 

26 

27 

28 

29. 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



83 513 
83 527 
83 540 
83 554 
83567 



83 580 
83 594 
83 607 
83 621 
83 634 



83 647 
83 661 
83 674 
83 688 
83 701 



83 714 
83 728 
83 741 
83 754 
83 768 



83 781 
83 794 
83 808 
83 821 
83 834 



83 847 
83 861 
83 874 
83 887 
83 900 



83 914 
83 927 
83 940 
83 953 
83 967 



83 980 

83 993 

84 006 
84 019 
84 033 



84 046 
84 059 
84 072 
84 085 
84 098 



84 111 
84 124 
84 138 
84 151 
84 164 



84 177 
Log. Cos. 



Log. Tan. c. d 



96 965 

96 991 

97 016 
97 041 
97 067 



97 092 
97 117 
97 143 
97 168 
97 193 



97 219 
97 244 
97 269 
97 295 
97 320 



97 345 
97 370 
97 396 
97 421 
97 446 



97 472 
97 497 
97 522 
97 548 
97 573 



97 598 
97 624 
97 649 
97 674 
97 699 



97 725 
97 750 
97 775 
97 801 
97 826 



97 851 
97 877 
97 902 
97 927 
97 952 



97 978 

98 003 
98 028 
98 054 
98 079 



98 104 
98 129 
98 155 
98 180 
98 205 



98 231 
98 256 
98 281 
98 306 
98 332 



98 357 
98 382 
98 408 
98 433 
98 458 



Log. Cot. Log. Cos 



9-98 483 
Log. Cot. 



003 
03 



034 
009 
984 
958 
933 
908 
882 
857 
832 
806 
781 
756 
730 
705 
680 



654 
629 
604 
578 
553 
528 
502 
477 
452 
427 



02 401 9 

02 

02 

02 

02 



: 376 

! 351 

I 325 

300 



275 
249 
224 
199 

m 

148 
123 
098 
072 
047 



022 
996 
971 
946 

921 



895 
870 
845 
819 
794 




642 
617 
592 
567 
54l 



01 516 



Log. Tan 



86 412 
86 401 
86 389 
86 377 
86 365 



86 354 
86 342 
86 330 
86 318 
86 306 



86 294 
86 282 
86 271 
86 259 
86 247 



86 235 
86 223 
86 211 
86 199 
86 187 



86 176 
86 164 
86 152 
86 140 
86 128 



86 116 
86 104 
86 092 
86 080 
86 068 



86 056 
86 044 
86 032 
86 020 
86 008 



85 996 
85 984 
85 972 
85 960 
85 948 



85 936 
85 924 
85 912 
85 900 
B5 887 



85 875 
85 863 
85 851 
85 839 
85 827 



85 815 
85 803 
85 791 
85 778 
85 766 



85 754 
85 742 
85 730 
85 718 
85 705 



9-85 693 
Log. Sin. 



60 

59 
58 

57 
56_ 
55 
54 
53 
52 
51 

50 

49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 

19 
18 
17 
16 
15 
14 
13 
12 
JA 

10 

9 
8 

7 
6 
5 
4 
3 
2 

O 



P. P. 



25. 



2 


5 


2. 


3 





2. 


3 


4 


3 


3 


8 


3 


4 


2 


4 


8 


5 


8 


12 


7 


12 


17 





16 


21 


2 


20 



25 

5 
9 
3 
7 
1 
3 
5 
6 
8 





13 


13 


6 


1-3 


1-3 


7 


1 


6 


1 


5 


8 


1 


8 


1 


7 


9 


2 





1 


9 


10 


2 


2 


2 


1 


20 


4 


5 


4 


3 


30 


6 


7 


6 


5 


40 


9 





8 




50 


11 


2 


10 


8 





12 


12 




6 


1.2 


1.2 




7 


1 


4 


1 


4 




8 


1 


6 


1 


6 




9 


1 


9 


1 


8 




10 


2 


1 


2 







20 


4 


1 


4 





3 


30 


6 


2 


6 


0'5 


40 


8 


3 


8 


0!7 


50 


10 


4 


10 





9. 



P.P. 



133° 



635 



46" 



44° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



135= 



10 

11 
12 
13 
14 

15 
16 
17 
18 
19 



20 

21 
22 
23 
21 
25 
26 
27 
28 
21 
30 
31 
32 
33 
3± 
35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59. 
60 



Log. Sin. d. Log. Tan. c.d. Log. Cot. Log. Cos.! d 



84 177 
84 190 
84 203 
84 216 
84 229 



84 242 
84 255 
84 268 
84 281 
84 294 

84 307 
84 320 
84 333 
84 346 
84 359 
84 372 
84 385 
84 398 
84 411 
84 424 



84 437 
84 450 
84 463 
84 476 
84 489 



84 502 
84 514 
84 527 
84 540 
84 553 
84 566 
84 579 
84 592 
84 604 
84 617 



84 630 
84 643 
84 656 
84 669 
84 681 



84 694 
84 707 
84 720 
84 732 
84 745 



84 758 
84 771 
84 783 
84 796 
84 809 



84 822 
84 834 
84 847 
84 860 
84 872 



84 885 
84 898 
84 910 
84 923 
84 936 



84 948 
Log. Cos. 



00 



Log. 



483 
509 
534 
559 
585 
610 
635 
660 
686 
711 
736 
762 
787 
812 
837 
863 
888 
913 
938 
964 

989 
014 
040 
065 
090 
115 
141 
166 
191 
216 
242 
267 
292 
318 
343 
368 
393 
419 
444 
469 
494 
520 
545 
570 
595 
621 
646 
67l 
697 
122 

747 
772 
798 
823 
848 
873 
899 
924 
949 
974 
000 
Cot, 



01 516 
01 491 
01 465 
01 440 
01 415 



01 390 
01 364 
01 339 
01 314 
01 289 



01 263 
01 238 
01 213 
01 187 
01 162 

01 137 
01 112 
01 086 
01 061 
01 036 



01 010 
00 985 
00 960 
00 935 
00J09 
00 884 
00 859 
00 834 
00 808 
00 783 
00 758 
00 733 
00 707 
00 682 
00 657 
00 631 
00 606 
00 581 
00 556 
00 530 



00 505 
00 480 
00 455 
00 429 
00 404 



00 379 
00 353 
00 328 
00 303 
00 278 



00 252 
00 227 
00 202 
00 177 
00 15l 



00 126 
00 101 
00 076 
00 050 
00 025 



0-00 000 
Log, Tan. 



85 693 
85 681 
85 669 



85 657 
85 644 



12 

12 

i 12 

7I 12 



85 632 
85 620 
85 608 
85 595 
85 583 



85 571 
85 559 
85 546 
85 534 
85 522 



85 509 
85 497 
85 485 
85 472 
85 460 



85 448 
85 435 
85 423 
85 411 
85 398 
85 386 
85 374 
85 361 
85 34? 
85 336 



85 324 
85 312 
85 299 
85 287 
85 274 



85 262 
85 249 
85 237 
85 224 
85 212 



85 199 
85 187 
85 174 
85 162 
85 149 



85 137 
85 124 
85 112 
85 099 
85 087 



85 074 
85 062 
85 049 
85 037 
85 024 



85 Oil 
84 99? 
84 986 
84 974 
84 961 



84 948 



il2 
12 
12 
12 
12 
12 
12 

I 12 
12 
12 
12 
12 
12 
12 
12 

I 12 
12 
12 
12 
12 

12 

' 12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 

12 
12 
12 
12 

12 
12 
12 
12 
13 

12 
12 
12 
12 
13 
12 



134* 



Log. Sin. d 
^36 



P. p. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



25 



2 


5 


2 


3 





2 


3 


4 


3 


3 


8 


3 


4 


2 


4 


8 


5 


8. 


12 


7 


12. 


17 





16. 


21 


2 


20. 



25 

5 
? 
3 
7 
I 
3 
5 
6 



13 



1 


3 


1 


1 


6 


1 


1 


8 


1 


2 





1 


2 


2 


2 


4 


5 


4 


6 


7 


6 


9 





8 


11 


2 


10 



13 

3 
5 
7 
p 

i 
3 
5 

6 
8 





1^. 


12 


6 


l.S 


1.2 


7 


1 


4 


1 


4 


e 


1 


6 


1 


6 


9 


1 


9 


1 


8 


10 


2 


1 


2 





20 


4 


1 


4 





30 


6 


2 


6 





40 


8 


3 


8 





50 


10 


4 


10 






P. p. 



45^ 



TABLE VIII. 

LOGARITHMIC VERSED SINES AND EXTERNAL 

SECANTS. 

637 



TABLE VTII.— LOGARITHMIC VERSED SINES AND EXTERNAL 
0° SECANTS. 1° 



Log. Vers. 



62642 
22848 
58066 
83054 



02436 
18272 
31662 
43260 
53490 



62642 
70920 
78478 
85431 
91868 



97860 
03466 
08732 
13696 
18393 



22848 
27086 
31126 
34987 
38684 



42230 
45636 
48915 
52073 
55121 



58066 
60914 
63672 
66344 
68937 



71455 
73902 
76282 
78598 
80854 



83053 
85198 
87291 
89335 
91332 



93284 
95193 
97061 
98890 
00680 



02435 
04155 
05842 
07496 
09120 



10714 
12279 
13816 
15327 
16811 



618271 
Log. Vers, 



60206 

35218 

24987 

19382 

15836 

13389 

11598 

10230 

9151 

8278 

7558 

6953 

6437 

5992 

5605 

5266 

4964 

4696 

4455 

4238 

4040 

3861 

3697 

3545 

3406 

3278 

3158 

3048 

2944 
2848 
2757 
2672 
2593 
2518 
2447 
2379 
2316 
2256 
2199 
2145 
2093 
2044 
1996 
1952 
1909 
1868 
1829 
1790 

1755 
1720 
1686 
1654 
1623 
1594 
1565 
1537 
1511 
1484 
1460 



Log. Exsec. 




62642 



97861 
03466 
08732 
13697 
18393, 



22849 
27087 
31127 
34988 
38685 



42231 
45638 
48916 
52075 
55123 



58068 
60916 
63674 
66346 
68940 



71457 
73904 
76284 
78601 
80857 



83056 
85201 
87295 
89338 
91335 
93288 
95197 
97065 
98894 
00685 



02440 
04160 
05847 
07501 
09125 



10719 
12284 
13822 
15333 
16818 



618278 
Log. Exsec, 



I> 



60206 

35218 

24987 

19382 

15836 

13389 

11598 

10230 

9151 

8279 

755^ 

6952 

6437 

5993 

5605 

5266 

4964 

4696 

4456 

4238 

4040 

3861 

3697 

3545 
3407 
3278 
3159 
3048 
2945 
2848 
2758 
2672 
2593 

2517 
2447 
2380 
2316 
2256 
2199 
2145 
2093 
2043 
1997 
1952 
1909 
1868 
1829 
1791 

1755 
1720 
1687 
1654 
1623 
1594 
1565 
1537 
1511 
1485 
1460 



Log. Vers. 



6. 18271 
19707 
21119 
22509 
23877 



25223 
26549 
27856 
29142 
30410 



31660 
32892 
34107 
35305 
36487 



37653 
38803 
39938 
41059 
42165 



43258 
44337 
45403 
46455 
47496 
48524 
49539 
50544 
51536 
52518 



53488 
54448 
55397 
56336 
57265 



58184 
59093 
59993 
60884 
61766 



62639 
63503 
6435? 
65206 
66045 
66876 
67700 
68515 
69323 
70124 



70917 
71703 
72482 
73254 
74019 



74777 
75529 
76275 
77014 
77747 



6-78474 
Log. Vers. 



2> 



1435 

1412 

1389 

1368 

1346 

1326 

1306 

1286 

1268 

1250 

1232 

1214 

1198 

1182 

1166 

1150 

1135 

1121 

1106 

1093 

1078 

1066 

1052 

1040 

1028 

1016 

1004 

992 

981 

970 

960 

949 

939 

929 

919 

909 

900 

891 

882 

872 

864 

855 

847 

839 

831 
823 
815 
808 
800 
793 
786 
779 
772 
765 
758 
752 
745 
739 
733 
726 



Log. Exsec, 



6. 18278 
19714 
21126 
22516 
23884 
25231 
26557 
27864 
29151 
30419 



31669 
32901 
34116 
35315 
36497 



37663 
38814 
3994? 
41070 
42177 



43270 
44349 
45415 
46468 
47509 



48537 
49553 
50557 
51550 
52532 



53503 
54463 
55413 
56352 
57281 



58201 
59110 
60011 
60902 
61784 



62657 
63522 
64378 
65226 
66065 



66897 
67725 
68536 
69345 
70145 



7093? 
71725 
72505 
73277 
74043 



74802 
75554 
76300 
77040 
77773 



6-78500 
Log. Exsec 



2> 



n 



638 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL 

2° SECANTS. 3° 



' Log. Vers. 



D 



Log. Exsec, 



Log. Vers. 



Log. Exsec, 



78474 
79195 
79909 
80618 
8132^ 
82019 
82711 
83398 
84079 
84755 



85425 
86091 
86751 
87407 
88057 



88703 
89344 



90612 
91239 



91862 
92480 
93093 
93703 
94308 



94909 
95506 
96099 
96688 
97272 



97853 
98430 
99004 
99573 
00139 



00701 
01259 
01814 
02366 
02914 



03458 
03999 
04537 
05071 
05603 



0613^) 
06655 
07177 
07695 
08211 



08723 
09232 
09739 
10242 
10743 



11240 
11735 
12227 
12716 
13203 



13687 



721 
714 
709 
703 
697 
692 
686 
681 
676 
670 
665 
660 
655 
650 

646 
641 
636 
631 
627 
622 
618 
613 
609 
605 

601 
597 
592 
58? 
584 

581 
577 
573 
569 
565 

562 
558 
555 
551 
548 

545 
541 
537 
534 
53l 
527 
525 
521 
518 
515 
512 
509 
506 
503 
500 
497 
495 
492 
48? 
486 
484 



78500 
79221 
79937 
80646 
81350 
82048 
82740 
83427 
8410? 
84785 



85457 
86123 
86783 
8743? 
88090 



88737 
89378 
90015 
90647 
91275 



91898 
92516 
93131 
93741 
94346 
94948 
95545 
96139 
96728 
97313 
97895 
98472 
99046 
99616 
00182 



00745 
01304 
01860 
02412 
02960 



03505 
04047 
04585 
05120 
05652 



06180 
06706 
07228 
07747 
08263 



08776 
09286 
09793 
10297 
10798 



11297 
11792 
12285 
12775 
1 3262 
13746 



721 
715 
70? 
703 
698 
692 
687 
682 
676 

671 
666 
660 
656 
651 

646 
641 
636 
632 
628 
623 
618 
614 
610 
605 
601 
597 
595 
58? 
585 
581 
577 
574 
570 
566 

563 
55? 
555 
552 
548 

545 
541 
538 
535 
531 

528 
525 
522 
519 
516 
513 
509 
507 
503 
501 

498 
495 
493 
490 
487 
485 



13687 
14168 
14646 
15122 
15595 



16066 
16534 
17000 
17463 
17923 



18382 
18837 
19291 
19742 
20191 
20637 
21081 
21523 
21963 
22400 



22836 
23269 
23700 
2412? 
24555 



24980 
25402 
25823 
26241 
26658 



27072 
27485 
27895 
28304 
28711 



29116 
29518 
29919 
3031? 
30716 



31112 
31505 
31897 
32288 
32676 



33063 
33448 
33831 

34213 
34593 



34971 
35348 
35723 
36097 
36468 



3683? 
37207 
37574 
37940 
38304 



38667 



481 
478 
475 
473 
470 
468 
466 
463 
460 

458 
455 
453 
451 
448 
446 
444 
442 
440 
437 
435 
433 
431 
42? 
426 
424 
422 
420 
418 
416 
414 
412 
410 
40? 
406 

405 
402 
401 
39? 
397 

395 
393 
392 
390 
388 

386 
385 
383 
382 
380 

378 
377 
375 
373 
-371 

370 
368 
367 
366 
364 
362 



7. 13746 
14228 
14707 
15183 
15657 



16129 
16598 
17064 
17528 
17989 



18448 
18905 
19359 
19811 
20260 



20707 
21152 
21595 
22035 
22473 



22909 
23343 
23775 
24204 
24632 



25057 
25480 
25902 
26321 
26738 



27153 
27567 
27978 
28387 
28795 



29200 
29604 
30006 
30406 
30804 



31201 
31595 
31988 
3237? 
32768 



33156 
33542 
33926 
3430? 
34689 



3506? 
35446 
35822 
36196 
36569 



36940 
37310 
37678 
38044 
38409 



38773 



481 
47? 
476 
474 
47l 
469 
466 
464 
46l 
45? 
456 
454 
452 
449 

447 
445 
442 
440 
438 
436 
434 
431 
42? 
427 
425 
423 
42l 
41? 
417 
415 
413 
411 
40? 
407 
405 
404 
402 
400 
398 
396 
394 
393 
391 
389 
388 
385 
384 
382 
380 

37? 
377 
376 
374 
373 
371 
369 
368 
366 
365 
363 



Log. Vers. 



2> 



Log. Exsec. 



J> 



Log. Vers. 



n 



Log. Exsec, 



z> 



639 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 

4° 5° 



Lg. Vers, 




42211 
42557 
42903 
43246 
43589 



43930 
44270 
44608 
44946 
45281 



45616 
45949 
46281 
46612 
M941 
47270 
47597 
47922 
48247 
48570 

48892 

49213 
49533 
49852 
50169 



50485 
50800 
51114 
51427 
51739 




56580 
56873 
57166 
57458 
57749 



7 58039 
Lg, Vers. 



361 
359 
358 
356 
355 
353 
352 
350 
349 
348 
346 
345 
343 
342 
341 
339 
338 
337 
335 
334 
333 
332 
330 
329 
328 
327 
325 
324 
323 
322 
321 
320 
318 
317 
316 
315 
314 
313 
311 
311 
309 
308 
307 
306 

305 
304 
303 
302 
300 
300 
299 
297 
297 
295 

295 
293 
293 
292 
290 
290 



Log.'Exs, 



7. 38773 
39134 
39495 
39854 
40211 



40567 
40922 
41275 
41627 
41977 



42326 
42673 
43019 
43364 
43708 



44050 
44390 
44730 
45068 
45405 



45740 
46075 
46407 
46739 
47070 



47399 
47727 
48054 
48379 
48703 



49026 
49348 
49669 
49989 
50307 



50624 
50941 
51256 
51569 
51882 



52194 
52504 
52814 
53122 
53429 



53735 
54041 
54345 
54648 
54950 



55251 
55550 
55849 
56147 
56444 



56740 
57035 
57329 
57621 
57913 



58204 
Log. Exs, 



361 
360 
359 
357 
356 
354 
353 
352 
350 
349 
347 
346 
345 
343 
342 
340 
339 
338 
337 

335 
334 
332 
332 
330 
329 
328 
327 
325 
324 

323 
322 
321 
319 
318 
317 
316 
315 
313 
313 

311 
310 
309 
308 
307 
306 
305 
304 
303 
302 

301 
299 
299 
298 
296 
296 
295 
294 
292 
292 
291 



Lg. Vers, 



58039 
58328 
58615 
58902 
59188 



59473 
59758 
60041 
60323 
6 0604 
60885 
61164 
61443 
61721 
61998 



62274 
62549 
62823 
63096 
63369 



63641 
63911 
64181 
64451 
647JJ 
64986 
65253 
65519 
65784 
66048 



66311 
66574 
66836 
67097 
67357 



67617 
67875 
68133 
68390 
68647 



68902 
69157 
69411 
69665 
_699r7 
70169 
70421 
70671 
70921 
71170 



71418 
71666 
71913 
72159 
72404 



72649 
72893 
73137 
73379 
73621 

73863 



Lg. Vers, 



i> Log.Exs. D 



289 
287 
287 
286 
285 
284 
283 
282 
281 
280 
279 
279 
277 
277 
276 
275 
274 
273 
272 

272 
270 
270 
269 
268 

267 
266 
266 
265 
264 
263 
263 
261 
261 
260 

259 
258 
258 
257 
256 
255 
255 
254 
253 
252 
252 
251 
250 
25C 
249 
248 
247 
247 
246 
245 
245 
244 
243 
242 
242 

241 



I) 



58204 
58494 
58783 
59071 
59358 



59645 
59930 
60214 
60498 
6078^ 
61062 
61342 
61622 
61901 
62179 



62456 
62733 
63008 
63282 
63556 



63829 
64101 
64372 
64643 
64912 
65181 
65449 
65716 
65982 
66247 



66512 
66776 
67039 
67301 
67562 
67823 
68083 
68342 
68601 
68858 



69115 
69371 
69627 
69881 
70135 



70388 
70641 
70893 
71144 
71394 



71644 
71892 
72141 
72388 
72635 



72881 
73126 
73371 
73615 
73859 
74101 



Log.Exs. 



290 
289 
288 
287 
286 
285 
284 
283 
282 

281 
280 
280 
279 
278 

277 
276 
275 
274 
274 
273 
272 
271 
270 
269 
269 
268 
267 
266 
265 
264 
264 
263 
262 
261 
261 
260 
259 
258 
257 
257 
256 
255 
254 
254 

253 
252 
252 
251 
250 
250 
248 
248 
247 
246 

246 
245 
245 
244 
243 
242 



P. P. 





360 


350 


34 


6 


360 


35-0 


34 


7 


42 





40 


fe 


39 


8 


48 





46 


6 


45 


9 


54 





51 


5 


51 


10 


60 





58 


3 


56 


20 


120 





116 


5 


113 


30 


180 





175 





170 


40 


240 





233 


3 


226 


50 


300 





291 


6 


283 





330 


320 


310 


6 


330 


32. 


31. 


7 


38 


5 


37 


3 


36 


1 


8 


44 





42 


6 


41 


3 


9 


49 


5 


48 





46 


5 


10 


55 





53 


3 


51 


6 


20 


110 





106 


6 


103 


3 


30 


165 





160 





155 





40 


220 





213 


3 


206 


6 


50 


275 





266 


6 


258 


3 





300 


290 


280 


6 


30.0 


29-0 


28. 


7 


35 





33 


3 


32 


6 


8 


40 





38 


5 


37 


3 


9 


45 





43 


5 


42 





10 


50 





48 


3 


46 


6 


20 


100 





96 


6 


93 


3 


30 


150 





145 





140 





40 


200 





193 


3 


186 


6 


50 


250 





241 


6 


233 


3 





27( 


[) 


260 


25 


6 


27-0 


260 


25 


7 


31 


5 


30 


3 


29 


8 


36 





34 


6 


33 


9 


40 


5 


39 





37 


10 


45 





43 


3 


41 


20 


90 





86 


6 


83 


30 


135 





130 





125 


40 


180 





173 


3 


166 


50 


225 





216 


6 


208 





240 


230 


220 


6 


24.0 


23. 


220 


7 


28 





26 


3 


25 


6 


8 


32 





30 


6 


29 


3 


9 


36 





34 


5 


33 





10 


40 





38 


3 


36 


5 


20 


80 





76 


6 


73 


3 


30 


120 





115 





110 





40 


160 





153 


3 


146 


Q 


50 


200 





191 


6 


183 


3 



10 
20 
30 
40 
50 



210 


200 


190 


210 


20. 


19. 


24 


5 


23 


3 


22 


1 


28 





26 


6 


25 


3 


31 


5 


30 





28 


5 


35 





33 


3 


31 


6 


70 





66 


6 


63 


3 


105 





100 





95 





140 





133 


3 


126 


6 


175 





166 


6 


158 


3 



P. p. 



640 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



Lg. Vers 



7. 73863 
74104 
74344 
74583 
74822 



75060 
75297 
75534 
75770 
76006 



76240 
76475 
76708 
76941 
77173 



77405 
77638 
77867 
78097 
78326 



78554 
78783 
79010 
79237 
79463 



79689 
79914 
80138 
80362 
80586 



80808 
81031 
81252 
81473 
81694 



I^ Log.Exs. I>', Lg. Vers 



81914 
82133 
82352 
82570 
82788 




85147 
85359 
85570 
85780 
85990 



86199 
86408 
86616 
86824 
87031 



87238 
Lg. Vers 



241 
240 
239 
239 

238 
237 
236 
236 
235 
234 
234 
233 
233 
232 
232 
231 
230 
230 
229 

228 
228 
227 
227 
226 
225 
225 
224 
224 
223 
222 
222 
221 
221 
220 
220 
219 
219 
218 
217 
217 
217 
216 
215 
215 
214 
214 
213 
213 
212 
212 
211 
211 
210 
210 
209 
209 
208 
208 
207 
206| 



74101 
74343 
74585 
74826 
75066 



75305 
75544 
75782 
76019 
76256 



76492 
76728 
76963 
77197 
77431 



77664 
77897 
78128 
78360 
78590 



78820 
79050 
79279 
79507 
79735 



79962 
80188 
80414 
80639 
80864 



81088 
81312 
81535 
81758 
81980 



82201 
82422 
82642 
82862 
83081 




85457 
85670 
85882 
86094 
86305 



86516 
86726 
86936 
87146 
87354 
87"563 



Log.Exs 



242 
241 
241 
240 
239 
239 
238 
237 
237 
236 
235 
235 
234 
233 
233 
232 
231 
231 
230 
230 
229 
229 
228 
228 
227 
226 
226 
225 
225 
224 
224 
223 
222 
222 
221 
221 
220 
219 
219 
219 
218 
217 
217 
216 
216 
215 
215 
214 
213 
213 
213 
212 
211 
211 
211 
210 
210 
209 
208 
208 



87238 
87444 
87650 
87855 
88060 



88264 
88468 
88672 
88875 
89077 
89279 
89481 
89682 
89882 
90082 



90282 
90481 
90680 
90878 
91076 



91273 
91470 
91667 
91863 
92058 




94181 
94371 
94561 
94751 
94940 





I> Lg. Vers 



2> Log.Exs. » 



206 
205 
2u5 
204 
204 
204 
203 
203 
202 

202 

201 

201 

200 

200 

199 

199 

198 

198 

197 

197 

197 

196 

196 

195 

195 

195 

194 

194 

193 

193 

192 

192 

191 

191 

190 

190 

190 

189 

189 

189 

188 

187 

188 

187 

186 

186 

186 

185 

185 

184 

184 

184 

183 

183 

1 

182 

182 

182 

181 

181 

Id 



87563 
87771 
87978 
88185 
88391 



88597 
88803 
89008 
89212 
89416 



89620 
89823 
90025 
90228 
90429 



90630 
9083i 
91032 
91231 
91431 



91630 
91828 
92027 
92224 
92421 



92618 
92815 
93010 
93206 
93401 



93596 
93790 
93984 
94177 
94370 



94562 
94754 
94946 
95137 
95328 




97401 
97587 
97773 
97958 
98143 



98327 
98512 
98695 
98879 
99062 



99244 



Log.Exs. 



208 
207 
207 
206 
206 
205 
205 
204 
204 
203 
203 
202 
202 
201 
201 
201 
200 
199 
199 
199 
198 
198 
197 
197 
197 
196 
195 
195 
195 
195 
194 
194 
193 
193 
192 
192 
192 
191 
191 
190 
190 
189 
189 
188 
188 
188 
188 
187 
187 
186 
186 
185 
185 
184 
184 
184 
183 
183 
183 
182 
■^ 



P.P. 





180 


9 


6 


18. 


0-91 


7 


21 





1 


1 


8 


24 





1 


2 


9 


27 





1 


4 


10 


30 





1 


6 


20 


60 





3 


1 


30 


90 





4 


7 


40 


120 





8 


3 


5C 


150 





7 


9 



6 

7 

8 

9 

10 

20 

30 

40 

50 






8 


0-8 


0. 


1 





0-9 


0. 


1 


1 


1.0 


1. 


1 


3 


1.2 


1. 


1 


4 


1-3 


1. 


2 


8 


2.6 


2. 


4 


2 


40 


3 


5 


6 


5.3 


5. 


7 


1 


6-6 


6 





7 


6 


( 


6 


0.7 


0-6 





7 


08 





7 





8 


0.9 





8 





9 


1.0 


1 








10 


1.1 


1 


1 


1- 


20 


2-3 


2 


1 


2 


30 


3.5 


3 


2 


3- 


40 


4.6 


4 


3 


4. 


50 


5.8 


5 


4 


5 



610. 



7 
8 
9 
10 
20 
30 
40 
50 



0.6!0 



0.7 
0.8 
0.9 
1.8 
2.7 
3.6 
4 6 



5 4_ 

5;o.4 

0.5 
0-6 
0.7 
0.7 
1.5 
2-2 
3-0 
3-7 



9 

0-9 
1.0 
1.2 
1.3 
1.5 
3.0 
4.5 
6.0 
7-5 



7 
8 
9 






4 

0-4 





3 


3 




6 


0.3 


0.3 





7 


0.4 





3 


0. 


8 


0.4 





4 


0. 


9 


0.5 





4 


0. 


10 


0-6 





5 





20 


1.1 


1 





0. 


30 


1.7 


1 


5 


1. 


40 


2.3 


2 





1. 


50 


2.9 


2 


5 


2. 



1_ 

601 



0.2 
0.2 
0.2 
0.2 
0.5 
0.7 

1.5 



2 

2 0.2 
• 3j0.2 
.3 0.2 
.4'0.3 
•4,0.3 
.8 06 
.2 1.0 
.611.3 
.11.6 

O 

0-0 



p.p. 



641 



TABLE VIII.- 



-LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 
8° 9'' 





180 


17 


f) 


16 


6 


18.0 


17.0 


16- 


7 


21.0 


19 


8 


18. 


8 


24-0 


22 


6 


21. 


9 


27.0 


25 


5 


24. 


10 


30.0 


28 


3 


26. 


20 


60-0 


56 


6 


53 


30 


90.0 


85 





80 


40 


120.0 


113 


3 


106 


50 


150-0 


141 


6 


133 




642 



TABLE VHT.—LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
10° 11° 



Lg. Vers, i>|Log.Exs. ^> Lg. Vers. I> Log. Exs. I> 



8 18162 
18306 
18450 
18594 
18738 
18881 
19024 
19167 
19309 
19452 



19594 
19736 
19878 
20019 
20160 



20301 
20442 
20582 
20723 
20863 



21003 
21142 
21282 
21421 
21560 



21698 
21837 
21975 
22113 
22251 



22389 
22526 
22663 
22800 
22937 



23073 
23209 
23346 
23481 
23617 



23752 
23888 
24023 
24158 
24292 



24426 
24561 
24695 
24828 
24962 



25095 
25228 
25361 
25494 
25627 



25759 
25891 
26023 
26155 
26286 



8-26417 
Lg. Vers. 



144 
144 
144 
143 
143 
143 
142 
142 
142 
142 
142 
142 
141 
14l 
141 
140 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
138 
137 
138 
137 
137 
136 
137 
136 
136 
136 
135 
136 
135 
135 
135 
135 
134 
134 
134 
134 
133 
133 
133 
133 
133 
132 
133 
132 
132 
132 
132 
131 
131 



18827 
18973 
19120 
19266 
19411 



19557 
19702 
19847 
19992 
20137 



20281 
20425 
20569 
20713 
20857 



21000 
21143 
21286 
21428 
21571 



21713 
21855 
21996 
22138 

22279 



22420 
22561 
22701 
22842 
22982 



23122 
23262 
23401 
23540 
23679 



23818 
23957 
24095 
24234 
24372 



24509 
24647 
24784 
24922 
25059 



25195 
25332 
25468 
25604 
25740 



25876 
26012 
26147 
26282 
2641J7 
26552 
26686 
26821 
26955 
27089 



146 
146 
146 
145 
145 
145 
145 
145 
144 
144 
144 
144 
144 
143 
143 
143 
143 
142 
142 
142 
142 
14l 
14l 
141 
141 
140 
140 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
138 

137 
138 
137 
137 
137 



8-29002 
•29129 
•29256 
•29383 
•29510 



136 
136 
136 
136 
136 
136 
135 
135 
135 
135 
134 
134 
134 
134 
134 
134 



8 27223 
Log, Exs. I O 



8-26417 
26548 
26679 
26810 
26941 



8-27071 
•27201 
•27331 
•27461 
-27590 



8-27719 
•27849 
•27977 
•28106 
•28235 



8-28363 
28491 
28619 
28747 
28875 



8^29636 
.29763 
.29889 
.30015 
•30140 



8-30266 
-30391 
•30516 
•30642 
•30766 



8-30891 
31015 
31140 
31264 
31388 



8-31511 
31635! 
•31758 
-31882 
-32005 



8-32128 
-32250 
-32373 
-32495 
•32617 



8-32739 
-32861 
-32983 
-33104 
-33225 



8-33347 

33468 

33588 

-33709 

^829 

8-33950 

Lg. Vers 



131 
131 
131 
130 
130 
130 
130 
130 
129 
129 
129 
128 
129 
128 
128 
128 
128 
128 
127 
127 
127 
127 
127 
126 
126 
126 
126 
126 
125 
125 
125 
125 
125 
124 
124 
124 
124 
124 
124 
123 
124 
123 
123 
123 
123 
122 
122 
122 
122 

122 
122 
121 
12l 
121 
12l 
121 
120 
120 
120 
120 



27223 
27356 
27490 
27623 
27756 



27889 
28021 
28153 
28286 
28418 



28550 
28681 
28813 
28944 
29075 



29206 
29336 
29467 
29597 
29727 



29857 
29987 
30117 
30246 
30375 



30504 
30633 
30762 
30890 
31019 




32418 
32544 
32670 
32796 
32922 



33047 
33173 
33298 
33423 
33547 
33672 
33797 
33921 
34045 
34169 

34293 
34417 
34540 
34663 
34786 



34909 



-O |Log.Exs 
643 



133 
133 
133 
133 
133 
132 
132 
132 
132 
132 
131 
131 
131 
131 
131 
130 
130 
130 
130 

130 
130 
129 
129 
129 

129 

129 

128 

1 

128 

128 

128 

127 

127 

127 

127 

127 

127 

126 

126 

126 

126 

126 

126 

125 

125 

125 

125 

125 

124 

125 
124 
124 
124 
123 
124 
124 
123 
123 
123 
123 





1 
2 
3 
4 
5 
6 
7 
8 
9 

10 

11 
12 
13 
Ik 
15 
16 
17 
18 
19 

30 

21 
22 
23 
2± 

25 
26 
27 
28 
29_ 
30 
31 
32 
33 
H 
35 
36 
37 
38 
39. 
40 
41 
42 
43 
il 
45 
46 
47 
48 
ii 
50 
51 
52 
53 
5£ 
55 
56 
57 
58 
li 
60 



P. P, 



6! 

7| 

101 
20; 



130 

13-0 
15 



30; 65 
40i 86 
50108 



130 

12.0 
14^ 



16 
18 
20 
40 
60 
80 
100 



10 
20 
30 
40 
50 



4_ 4 

0-40-410 
-50 
-60 
-70 
-70 
-51 

• 2:2 

.0:2 

• 73 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 






3 








3 








4 








4 








5 


0- 


1 





0- 


1 


5 


1- 


2 





1- 


2 


5 


2. 






2 








2 








2 








3 








3 








6 





1 








1 


3 


1. 


1 


6 


1. 



0_ 

00 



p. p. 



TABLE VITl.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
12° 13° 




' Lg. Vers. J> Log.Exs. T) Lg. Vers. J> Log.Exs 



P. P. 





130 


119 


11 


6 


12-0 


11.9 


11. 


7 


14 





13 


9 


13. 


8 


16 





15 


8 


15. 


9 


18 





17 


8 


17. 


10 


20 





19 


8 


19. 


20 


40 





39 


G 


39. 


30 


60 





59 


5 


59. 


40 


80 





79 


3 


78- 


50 


100 





99 


1 


98. 





11 


7 


116 


6 


11.7 


11-61 


7 


13 


6 


13 


5 


8 


15 


6 


15 


4 


9 


17 


5 


17 


4 


10 


19 


5 


19 


3 


20 


39 





38 


6 


30 


58 


5 


58 





40 


78 





77 


3 


50 


97 


5 


96 


6 





114 


113 


6 


11.4 


11-31 


7 


13 


3 


13 


2 


8 


15 


2 


15 





9 


17 


1 


16 





10 


19 





18 


8 


20 


38 





37 


6 


30 


57 





56 


5 


40 


76 





75 


3 


50 


95 





94 


1 



111 

11.1 

12 



110 

11-0 
12 



115 

11.5 
13.4 
15-3 
17-2 
19-1 
383 
57-5 
76.6 
958 

113 

11-2 
13.0 
14-9 
16-8 
18-6 
373 
56-0 
74.6 
93-3 

109 

10-9 



12._ 
14-5 
_ 16-3 
3 18-1 
6 36-3 
54.5 
3 72.6 
6'90.8 



108 

6 10-8 

7 12 

8 14 

9 16 
10 18 
20 36 
30154 
40172 
50!90 



107 



105 


104 


10-5 


10. 4| 


12.2 


12 


1 


14-0 


13 


8 


15-7 


15 


6 


17.5 


17 


3 


35.0 


34 


6 


52-5 


52 





70-0 


69 


3 


87 -5 


86 


6 



106 

10-6 

12-3 

14-1 

15-9 

_ 17-6 

6 35-3 

5 53-0 

3 70-6 

1 883 



0.0 



iO.l 
► 0.2 
i 0-3 
;0.4 



P. P: 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
14° 15° 



Lg.Vers. -» Log.Exs. I> Lg. Vers. D Log.Exs. 2> 



47282 
47384 
47487 
47590 
47692 



47795 
47897 
47999 
48101 
48203 



48304 
48406 
48507 
48609 
48710 



48811 
48912 
49013 
49114 
49215 



49315 
49415 
49516 
49616 
49716 



49816 
49916 
50015 
50115 
50215 



50314 
50413 
50512 
50611 
50710 



50809 
50908 
51006 
51105 
5120^ 



51301 
51399 
51497 
51595 
51693 



51791 
51888 
51986 
52083 
52180 



52277 
52374 
52471 
52568 
52665 



52761 
52858 
52954 
53050 
53146 



53242 



' Lg.Vers. 



102 
103 
102 
102 
102 
102 
102 
102 
102 

101 

101 

101 

101 

101 

101 

101 

101 

100 

101 

100 

100 

100 

100 

100 

100 

100 

99 

100 

99 

99 

99 

99 

99 

99 

98 

9? 

98 

98 

98 

98 

98 

98 

98 

97 

98 

97 

97 

97 

97 

97 

97 

97 

96 

97 

96 

96 

96 

96 

96 

96 



48591 
48697 
48803 
48909 
49014 



49120 
49225 
49331 
49436 
49541 



49646 
49750 
49855 
49960 
50064 



50168 
50273 
50377 
50481 
50585 



50688 
50792 
50896 
50999 
51102 



51205 
51309 
51412 
51514 
51617 



51720 
51822 
51925 
52027 
52129 



52231 
52333 
52435 
52537 
5263P 



52740 
52841 
52943 
53044 
53145 



53246 
53347 
53448 
53548 
53649 



53749 
53850 
53950 
54050 
5 4150 
54250 
54350 
54449 
54549 
54649 



8.54748 
Log.Exs 



106 
106 
105 
105 
105 
105 
105 
105 
105 

105 
104 
105 
104 
104 
104 
104 
104 
104 
104 
103 
104 
103 
103 
103 
103 
103 
103 
102 
103 
102 
102 
102 
102 
102 
102 
102 
102 
101 
101 
101 
101 
lOl 
101 
101 
101 
101 
101 
100 
100 
100 
100 
100 
100 
100 
100 
100 

99 
100 

99 



53242 
53338 
53434 
53530 
53625 



53721 
53816 
53911 
54007 
54102 



54197 
54291 
54386 
54481 
54575 
54670 
54764 
54868 
54952 
55046 



55140 
55234 
55328 
5542i 
55515 



55608 
55701 
55795 
55888 
55981 



56074 
56166 
56259 
56352 
56444 



56536 
56629 
56721 
56813 
56905 



56997 
57089 
57180 
57272 
57363 



57455 
57546 
57637 
57728 
57819 



57910 
58001 
58092 
58182 
58273 



58363 
58453 
58544 
58634 
58724 
8-58814 
1> |Lg. Vers, 



8-54748 
.54847 
.54946 
.55045 
.55144 



8-55243 
.55342 
•55441 
-55539 
•55638 



8-55736 
.55834 
.55933 
.56031 
.56129 



8-56226 
.56324 
.56422 
.56519 
•56617 



8.56714 
.56812 
.56909 
.57006 
-57103 



8.57200 
-57296 
.57393 
.57490 

__-57586 



8-57682 
-57779 
-57875 
-57971 
•58067 



8.58163 
.58259 
.58354 
.58450 
•58546 



8-58641 
-58736 
.58832 
.58927 
-59022 



8-59117 
.59211 
.59306 
.59401 
•59495 



8-59590 
.59684 
.59779 
.59873 
•59967 



8.60061 
.60155 
.60249 
-60342 
.60436 



8-60530 



Log.Exs 



98 



P.P. 





103 


103 


6 


10-3 


10-2 


7 


12-0 


11-9 


8 


13-7 


13-6 


9 


15.4 


15-3 


10 


17-1 


17-0 


20 


34-3 


34.0 


30 


51-5 


51-0 


40 


68-6 


68-0 


50 


85-8 


85-0 



100 

6 10. 

711 

8 13 

9,15 
10 16 
20 33 
30 50 
40 66 
50183 



ICl 
11.8 
13-4 
15-1 
16^8 
33. 6 
50-5 
67-3 
84-1 



98 





97 


96 


6 


9-7 


9-6 


7 


11 


3 


11-2 


8 


12 


9 


12-8 


9 


14 


5 


14^4 


10 


16 


1 


16-0 


20 


32 


3 


32^0 


30 


48 


5 


48-0 


40 


64 


6 


64.0 


50 


80 


8 


80-0 



99 

9-91 
11-5 11-4 
13-2 13-0 
14-8114-7 
16-5|16-3 
33.0;32-6 
49.549.0 
86.0 65.3 
82.5182.6 



95 

9-5 
11.1 
12-6 
14-2 
15-8 
31-6 
47-5 
63-3 
79.1 



10 
20 
30 
40 
50 



94 

9.4 
10-9 
12.5 
14-1 
15-6 
31-3 
47-0 
62-6 
78-3 




91 


90 


9.1 


9^0 


10.6 


10^5 


12^1 


12^0 


13.6 


13^5 


15-1 


15.0 


30-3 


30.0 


45.5 


45.0 


60-6 


60.0 


75.8 


75-0 



o 

0.0 
0-0 
0-0 
0-1 
0.1 
O.I 
0.2 
0.3 
0.4 



P.P. 



645 



TABLE VIII.— LOGARITHMIC VKRSED SINES AND EXTERNAL SECANT& 
16° 17° 



Lg.Vers, J> Log.Exs. J> Lg. Vers. I> Log.Exs. -» 




60152 
60240 
60328 
60417 
60505 



60593 
60681 
60769 
60857 
60944 



61032 
61119 
61207 
61294 
61381 



61469 
61556 
61643 
61730 
61816 



61903 
61990 
62076 
62163 
62249 



62336 
62422 
62508 
62594 
62680 



62766 
62852 
62937 
63023 
63108 



63194 
63279 
63364 
63449 
63534 



63619 
63704 
63789 
63874 
63959 



8-64043 
Lg.Vers. 



60530 
60623 
60716 
60810 
60903 



60996 
61089 
61182 
61275 
61368 



61460 
61553 
61645 
61738 
61830 



61922 
62014 
62106 
62198 
62290 



62382 
62474 
62565 
62657 
62748 



62840 
62931 
63022 
63113 
63204 




64199 
64289 
64379 
64469 
64559 



64649 
64738 
64828 
64917 
65006 




8-65984 



Log.Exs. 



2) 



64043 
64128 
64212 
64296 
64381 



64465 
64549 
64633 
64717 
64801 



64884 
64968 
65052 
65135 
65218 



65302 
65385 
65468 
65551 
65634 



65717 
65800 
65883 
65965 
6604 8 
66131 
66213 
66295 
66378 
66460 



66542 
66624 
66706 
66788 
66870 



66951 
67033 
67115 
67196 
67277 



67359 
67440 
67521 
67602 
67683 




68969 



Lg. Ven 



2> 




65984 
66072 
66160 
66248 
66336 



66425 
66512 
66600 
66688 
66776 



67736 
67822 
67909 
67996 
68082 



68169 
68255 
68341 
68428 

68514 




69457 
69542 
69627 
69712 
69798 



69883 
69967 
70052 
70137 
70222 
70306 
70391 
70475 
70560 
70644 



70728 
70813 
70897 
70981 
71065 



8-71149 



Log.Exs. 



n 



o 

1 

2 

3 

_4 

5 

6 

7 

8 

_^ 

10 

11 

12 

13 

JA 

15 

16 

17 

18 

19 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 

11 
35 
36 
37 
38 
39 

40 
41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



P.P. 





93 


93 


9J 




6 


9.3 


9.2 


9.1 


7 


10.8 


10 


7 


10 


6 


8 


12.4 


12 


^ 


12 


1 


9 


13.9 


13 


8 


13 


6 


10 


15.5 


15 


3 


15 


1 


20 


31.0 


30 


6 


30 


5 


30 


46.5 


46 





45 


5 


40 


62.0 


61 


3 


60 


6 


50 


77.5 


76 


6 


75 


8 





90 


89 


8J= 


6 


9.0 


89 


8 


7 


10 


5 


10 


4 


10 


8 


12 





11 


8 


11 


9 


13 


5 


13 


3 


13 


10 


15 





14 


3 


14 


20 


30 





29 


g 


29 


30 


45 





44 


5 


44 


40 


60 





59 




58 


50 


75 





74 


1 


73 



87 86 85 

5 
? 
3 
7 
1 
3 
5 



8 


7 


8 


6 


8. 


10 


1 


10 





9 


11 


6 


11 


4 


11 


13 





12 


9 


12- 


14 


5 


14 


3 


14 


29 





28 


6 


28 


43 


5 


43 





42 


58 





57 


3 


56 


72 


5 


71 


6 


70 





84 


83 


82 


6 


84 


83 


8 


7 


9 


8 


9 


7 


9 


8 


11 


2 


11 





10 


9 


12 


6 


12 


4 


12 


10 


14 





13 


8 


13- 


20 


28 





27 


5 


27. 


30 


42 





41 


5 


41- 


40 


56 





55 


3 


54. 


50 


70 





69 


1 


68- 





81 


80 


79 


6 


8-1 


8-0 


7-9 


7 


9-4 


9 


3 


9 


2 


8 


10. 8 


10 


6 


10 


5 


9 


12-1 


12 





11 


g 


10 


13-5 


13 


3 


13 


1 


20 


27-0 


26 


6 


26 


3 


30 


40-5 


40 





39 


5 


40 


54.0 


53 


3 


52 




50 


67-5 


66 


6 


65 


8 



6 

7 

8 

9 

10 

20 

30 

40 



0% 

o.o 

0-0 
0.1 
0.1 
0.1 
0.2 
0-3 



P.P. 



(346 



^^ TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
18° 19° 



' Lg. Vers. ^> Log.Exs. X> Lg. Vers. I> Log. Exs. Z> 



8.68969 
69049 
69129 
69208 
69288 



69367 
69446 
69526 
69605 
69684 



69763 
69842 
69921 
70000 
70079 



70157 
70236 
70314 
70393 
70471 



70550 
70628 
70706 
70784 
70862 



70940 
71018 
71096 
71174 
71251 
71329 
71406 
71484 
71561 
71639 
71716 
71793 
71870 
71947 
72024 



72101 
72178 
72255 
72331 
72408 



72485 
72561 
72637 
72714 
72790 



72866 
72942 
73018 
73094 
73170 
73246 
73322 
73398 
73473 
73549 



8-73625 
Lg. Vers. 



72812 
72894 
72977 
73059 
73141 



73223 
73306 
73388 
73470 
73551 



73633 
73715 
73797 
73878 
73960 



74041 
74123 
74204 
74286 
743 B 7 



74448 
74529 
74610 
74691 
74772 

74853 
74934 
75014 
75095 
75175 



75256 
75336 
75417 
75497 
75577 
75658 
75738 
75818 
75898 
75978 



8-76058 
Log.Exs, 





.75121 
.75195 
.75269 
.75343 
.75417 



8. 



75491 
75565 
75639 
75712 
75786 



.75860 
•75933 
.76006 
.76080 
.76153 



.76226 
.76300 
.76373 
•76446 
.76519 



.76592 
.76664 
.76737 
.76810 
.76883 



8.76955 
•77028 
.77100 
.77173 
.77245 



8.77317 
.77390 
.77462 
.77534 
.77606 



8.77678 
.77750 
.77822 
.77893 
.77965 



8-78037 
Lg. Vers, 



76058 
76137 
76217 
76297 
76376 



76456 
76536 
76615 
76694 
76774 



76853 
76932 
77011 
77090 
77169 



77248 
77327 
77406 
77485 
77563 



77642 
77720 
77799 
77877 
77956 



78034 
78112 
78191 
78269 
78347 





79973 
80050 
80126 
80203 
80280 
80356 
80433 
80509 
80586 
80662 



80738 



X> Log.Exs, 
~647 





1 

2 
3 

4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
JL4 

15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39_ 
40 
41 
42 
43 
il 
45 
46 
47 
48 
4i 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59. 
60 



P.P. 





84 


83 


6 


8.4 


8.3 


7 


9.8 


9.7 


8 


11.2 


11.0 


9 


12.6 


12.4 


10 


14.0 


13.8 


20 


28.0 


27-6 


30 


42.0 


41.5 


40 


56.0 


55.3 


50 


70.0 


69.1 





81 


80 


6 


8.1 


8.0 


7 


9 


4 


9 


3 


8 


10 


8 


10 


6 


9 


12 


1 


12 





10 


13 


5 


13 


3 


20 


27 





26 


6 


30 


40 


5 


40 





40 


54 





53 


3 


50 


67 


5 


66 


6 





78 


77 


6 


7.8 


7.7 


7 


9.1 


9.0 


8 


10.4 


10.2 


9 


11.7 


11.5 


10 


13.0 


12.8 


20 


26.0 


25.6 


30 


39.0 


38.5 


40 


52.0 


51-3 


50 


65-0 


64-1 





75 


74 


6 


7.5 


7.4 


7 


8.7 


8.6 


8 


10.0 


9.8 


9 


11.2 


11.1 


10 


12.5 


12.3 


20 


25.0 


24.6 


30 


37.5 


37.0 


40 


50.0 


49.3 


50 


62.5 


61.6 



82 
8-2 
9.5 
10.9 
12.3 
13.6 
27.3 
41.0 
54.6 
68.3 



79 

7-9 
9.2 
10.5 
11.8 
13.1 
26-^ 
39.5 
52.6 
65.3 



76 

7.6 
8.8 

10.1 
11.4 
12.6 
25.3 
38.0 
50.6 
63-3 



73 

7-3 
8.5 

9.7 
10.9 
12.1 
24.3 
36.5 
48.6 
60.8 



10 
20 
30 
40 
50 



73 

7.2 

8.4 

9.6 

10.8 

12.0 

24-0 

36.0 

48.0 



60.0 59 



71 

7.1 
3 
4 

10-6 
11 " 
23 
35 
47 



5 

Q 

1 

1 

T 

0-2 

0-3 

0.4 



P.P. 



TABLE VIIT.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
30"* 21° 



' Lg. Vers. I> Log.Exs. 



n Lg.Vers. l>'Log.Exs. I> ' 




P.P. 




78750 
78821 
78892 
78963 
79034 



79105 
79175 
79246 
79317 
79387 



79458 
79528 
79598 
79669 
79739 



79809 
79879 
79949 
80019 
80089 



80159 
80229 
80299 
80369 
80438 
80508 
80577 
80647 
80716 
80786 



80855 
80924 
80993 
81063 
81132 



81201 
81270 
81339 
81407 
81476 



81545 
81614 
81682 
81751 
81819 




8-82229 
Lg.Vers 



/> 




81498 
81573 
81649 
81725 
81800 



81876 
81951 
82026 
82102 
82177 



82252 
82327 
82402 
82477 
82552 



82627 
82702 
82776 
82851 
82926 



83005 
83075 
83149 
83224 
83298 



83373 
83447 
83521 
83595 
83670 




84113 

84187 

8426 

8433 

84408 



84481 

84555 
8462§ 
84702 
8477 5 
84845 
84922 
84995 
85068 
85141 



8-85214 
Log.Exs., 




83246 
83313 
83380 
83447 
83515 



83582 
83649 
83716 
83783 
83850 

83916 
83983 
84050 
84117 
84183 



84250 
84316 
84383 
84449 
84515 



84582 
84648 
84714 
84780 
84846 



84912 
84978 
85044 
851ir 
8517i 



85242 
85308 
85373 
85439 
85505 



855^0 
85626 
85'70i 

85832 



85897 
85962 
86027 
86092 
86158 



86223 
Lg.Vers. 



8.85214 
.85287 
.85360 
.85433 
.85506 



86664 
86736 
86808 
8688(5 
86952 



87024 
87095 
87167 
87239 
87316 



87382 
87453 
87525 
87596 
87668 



87739 
87810 
87881 
87953 
88024 




88«>03 
888''3 
88944 
89014 
89085 



89155 
89225 
89295 
89366 
89436 



89506 



Log.Exs. 





1 
2 
3 

_4 
5 
6 
7 
8 

_9 

10 

11 
12 
13 
JLi 
15 
16 
17 
18 
, 19 

20 

21 
22 
23 
.24 
25 
26 
27 
28 
-29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 

41 
42 
43 
ii 
45 
46 
47 
48 
49 

50 

51 
52 
53 

55 
56 
57 
58 
59. 
60 





76 


75 


74 


6 


7.6 


7.5 


7. 


7 


8.8 


8.7 


8. 


8 


10.1 


10.0 


9. 


9 


11.4 


11.2 


11. 


10 


12.6 


12.5 


12. 


20 


25.3 


25-0 


21. 


30 


38.0 


37.5 


37. 


40 


50.6 


50. C 


49. 


50163.3 


62.5 


61. 



7-3 


7. 


8.5 


8. 


9.7 


9. 


10.9 


10 


12.1 


12 


24 3 


24 


36.5 


36 


48.6 


48. 


60-8 


60. 



\ 71 

2 7.1 
4 8.3 



70 

7.0 

1 



3 

5 10 

6 11 
3 23 
34 
6 46 
3 57 



8 10.6 
11.8 
023.6 
035.5 
047-5 
59.1 



69 68 

6.91 6.8 

8.0| 7.9 

9.2, 9.0 

3 10.2 

• 5,11.3 

.0'22.6 

5 34.0 

.045.3 

.5 56.6 



6 

7 

8 

9 

10 

20 

30 

40 

50 





67 


66 


65 


6 


6.7 


6.6 


6.5 


7 


7 


8 


7 


7 




8 


8 


9 


8 


8 


8.6 


9 


10 





9 


9 


9.7 


10 


11 


1 


11 





10.8 


20 


22 


3 


22 





21.6 


30 


33 


5 


33 





32.5 


40 


44 


6 


44 





43.3 
54.1 


50 


55 


8 


55 






P. p. 



648 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
33° 33° 



Lg.Vers. J> Log.Exs. i> Lg.Vers. 2> Log.Exs, 



8.86223 
86287 
86352 
86417 
86482 



86547 
86612 
86676 
86741 
86805 



86870 
86934 
86999 
87063 
87127 



87192 
87256 
87320 
87384 
87448 



87512 
87576 
87640 
87704 
87768 



87832 
87895 
87959 
88023 
88086 



88150 
88213 
88277 
88340 
88404 



88467 
88530 
88593 
88656 
88720 



88783 
88846 
88909 
88971 
89034 



S9097 
89160 
89223 
89285 
89348 



89411 
89473 
89536 
89598 
89660 



89723 
89785 
89847 
89910 
89972 



8-00034 
Lg. Vers, 



•89506 
.89576 
.89646 
.89716 
.89786 



.89856 
.89926 
•89995 
.90065 
.90135 



.90205 
.90274 
.90344 
.90413 
.90483 



.90552 
.90622 
.90691 
.90760 
.90830 



•90899 
•90968 
•91037 
•91106 
•91175 




.91932 
.92000 
.92068 
.92137 

.92205 



•92274 
.92342 
.92410 
.92478 
.92546 



8.92615 
.92683 
.92751 
.92819 
.92887 



8.92955 
.93022 
.93090 
•93158 
•93226 

8.93293 
.93361 
•93429 
.93496 
•93564 



8-93631 
Log.Exs 




90034 
90096 
90158 
90220 
90282 



91267 
91328 
91389 
91450 
91511 



91572 
91633 
91694 
91755 
91815 



91876 
91937 
91997 
92058 
92119 



92179 
92240 
92300 
92361 
92421 



92487 
92542 
92602 
92662 
92722 



92782 
92842 
92902 
92962 
93022 



93082 
93142 
93202 
93261 
93321 
93381 
93440 
93500 
93560 
93619 



93679 



Lg. Vers 



93631 
93699 
93766 
93833 
93901 



93968 
94035 
94102 
94170 
94237 



94304 
94371 
94438 
94505 
94572 



94638 
94705 
94772 
94839 
94905 



94972 
95039 
95105 
95172 
9 5238 
95305 
95371 
95437 
95504 
95570 




96297 
96362 
96428 
96494 
96560 



96625 
96691 
96757 
96822 
96888 



96953 
97018 
97084 
97149 
97214 



97280 
97345 
97410 
97475 
97540 



8-97606 



Log.Exs, 



P.P. 





70 


6 


7.0 


7 


8.1 


8 


9.3 


9 


10.5 


10 


11.6 


20 


23.3 


30 


35.0 


40 


46.6 


50 


58.3 



69 



6.9 


6. 


8.0 


7. 


9.2 


9^ 


10.3 


10. 


11.5 


11^ 


23.0 


22. 


34.5 


34 • 


46.0 


45 • 


57.5 


56. 



68 

8 



6 

7 

8 

9 

10 

20 

30 

40 

5o: 



67 66 



6.7 


6 


6 


6. 


7.8 


7 


7 


7- 


8.9 


8 


8 


8- 


10^0 


9 


9 


9- 


11.1 


11 





10. 


22^3 


22 





21. 


33^5 


33 





32- 


44-6 


44 





43. 


55.8 


55 





54 



64 63 



6 


4 


6 


3 


6. 


7 


4 


7 


3 


7- 


8 


5 


8 


4 


8- 


9 


6 


9 


4 


9- 


10 


6 


10 


5 


10- 


21 


3 


21 





20- 


32 





31 


5 


31- 


42 


6 


42 





41- 


53 


3 


52 


5 


51. 





61 


60 


59 


6 


6.1 


6-0 


5- 


7 


7.;. 


7 





6- 


8 


8. . 


8 





7- 


9 


9. 


9 





8- 


10 


10.: 


10 





9- 


20 


20-: 


20 





19- 


30 


30.5 


30 





29. 


40 


40.6 


40 





39. 


50 


50.8 


50 





49. 



65 

5 



63 

2 
2 
2 
3 
3 
6 

3 
6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



n 

0.1 

n 

0.4 



P.P. 



649 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
24° 25° 



Lg. Vers. J> Log.Exs. J> Lg.Vers. -£> Log.Exs. I> 



' 


Lg.Vers. 


2> 




1 

2 
3 
4 


8 


93679 
93738 
93797 
93857 
93916 


59 
59 
59 
59 

59 
59 
59 
59 
59 
59 
59 
59 
59 
58 

59 
59 
58 
59 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
58 
57 
58 
58 
58 
57 
57 
58 
57 
57 
57 
57 
57 
57 
57 
57 

^1 

57 
57 
57 

1? 

57 
57 


5 
6 
7 
8 
9 


8 


93975 
94034 
94094 
94153 
94212 


10 

11 
12 
13 
14 


8 


94271 
94330 
94389 
94448 
94506 


15 
16 
17 
18 
19 


8 


94505 
94624 
94683 
94742 
94800 


20 

21 
22 
23 
24 


8 


94859 
94917 
94976 
95034 
95093 


25 
26 
27 
28 
29 


8 


95151 
95210 
95268 
95326 
95384 


30 

31 
32 
33 
34 


8 


95443 
95501 
95559 
95617 
95675 


35 
36 
37 
38 
39 


8 


95733 
95791 
95849 
95907 
95965 


40 

41 
42 
43 
44 


8 


96023 
96080 
96138 
96196 
96253 


45 
46 
47 
48 
49 

50 

51 
52 
53 
54 


8 
8^ 


96311 
96368 
96426 
96483 
96541 

96598 
96656 
96713 
96770 
96827 


55 
56 
57 
58 
59 


8 


96885 
96942 
96999 
97056 
97113 


60 


8 


97170 


' 


U 


j.Vers. 


D 



97606 
97671 
97736 
97801 
97865 



97930 
97995 
98060 
98125 
98190 



98254 
98319 
98383 
98448 
98513 



98577 
98642 
98706 
98770 
98835 



98899 
98963 
99028 
99092 
99156 



99220 
99284 
99348 
99412 
99476 



99540 
99604 
99668 
99732 
99796 
99860 
99923 
99987 
00051 
00114 



00178 

0024 

00305 

00369 

00432 

00495 

00559 

00622 

00686 

00749 

00Tl2 

00875 

00938 

010C2 

01065 



01128 
01191 
01254 
01317 
01380 



9-01443 



Log.Exs. 



2> 




97738 
97795 
97851 
97908 
97964 



98020 
98077 
98133 
98190 
98246 



98302 
98358 
98414 
98470 
98527 



98583 
98639 
98695 
98750 
98806 



98802 
98918 
98974 
99030 
99085 



99141 
99197 
99252 
99308 
99363 



99419 
99474 
99529 
99585 
99640 



99695 
99751 
99806 
99861 
99916 



99971 
00026 
00081 
00136 
00191 
0024g 
00301 
00356 
00411 
00466 



9.00520 
Lg. Vers. 



9-01443 
01505 
01568 
01631 
01694 



01756 
01819 
01882 
01944 
02007 



02070 
02132 
02195 
02257 
02319 



02382 
02444 
02506 
02569 
02631 



02693 
02755 
02817 
0288C 
02942 




03622 
03684 
03746 
03807 
03869 



03930 
03992 
04053 
04115 
04176 




9-05154 



Log.Exs. 



P. P. 



65 

6.5 



64 

6-4 

6' 7-4 

6i 851 8 

7! 9.6 9 

8110. 6110 



6,21.3 
532.0 
3 42.6 
1I53.3 



63 

6-3 
3 
4 
4 
5 

5 

5 



10 
20 
30 
40 
50 



62 

6-2 

7-2 

8-2 

9-3 

10.3 

20-6 

31-0 

43.3 

51.6 



61 

6-1 



60 

6.0 





56 


55 


6 


5.6 


5-5! 


7 


6 


5 


6 


4 


8 


7 


4 


7 


3 


9 


8 


4 


8 


2 


10 


9 


3 


9 


1 


20 


18 


6 


18 


3 


30 


28 





27 


5 


40 


37 


3 


36 


6 


50 


46 


6 


45 


8 



54 

5.4 

6.3 

7.2 

8.1 

9.0 

18.0 

27.0 

36.0 

45.0 



O 

0.0 
0.0 
0.0 
0.1 
0.1 
O.T 
0.2 
0.3 
0.4 



P. P. 



59 


58 


57 


5.9 


5-8 


5.7 


6 


9 


6 


7 


6.6 


7 


8 


7 


7 


7.6 


8 


8 


8 


7 


8.5 


9 


8 


9 


6 


9.5 


19 


6 


19 


3 


19.0 


29 


5 


29 





28.5 


39 


3 


38 


6 


38.0 


49 


1 


48 


3 


47.5 



650 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANXa 
26° 27° 




P.P. 





61 


60 


6 


6.1 


6.0 


7 


7.1 


7.0 


8 


8.1 


8-0 


9 


9.1 


9.0 


10 


10.1 


10.0 


20 


20.3 


20.0 


30 


30.5 


30.0 


40 


40.6 


40.0 


50 


50.8 


50.0 





58 


6 


58 


7 


6.7 


8 


7.7 


9 


8.7 


10 


9.6 


20 


19.3 


30 


29.0 


40 


38.6 


50 


48-3 





55 


6 


5.5 


7 


6.4 


8 


7.3 


9 


8.2 


10 


9.1 


20 


18. 3 


30 


27.5 


40 


36.6 


50 


45.8 



54 

5.4 
6.3 
7.8 

8.1 

S.Q 
18.0 
27.0 
36.0 
43.0 




P. P. 



651 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT.' 
28° 39° 



' Lg. Vers. I> Log.Exs. X> Lg. Vers. I> Log.Exs. l> 



06838 
06888 
06939 
06990 
07040 



07091 
07141 
07192 
07242 
07293 



07343 
97393 
07444 
07494 
07544 



07594 
07644 
07695 
07745 
07795 



07845 
07895 
07945 
07995 
08045 
08095 
08145 
08195 
0C244 
08294 



08344 
08394 
08443 
08493 
08543 



08592 
08642 
08691 
08741 
08790 



08840 
08889 
08939 
08988 
09087 



09087 
09136 
09185 
09234 
09284 



09333 
09382 
09431 
09480 
09529 



09578 
09627 
09676 
09725 
09774 



909R93 
Lg. Vers. 



9.12244 
12302 
12359 
12416 
12474 



12531 
12588 
12645 
12703 
1 2760 

12817 
12874 
12931 
12988 
13045 



13102 
13159 
13216 
13273 
13330 



13387 
13444 
13500 
13557 
13614 



13671 
13727 
13784 
13841 
13897 



13954 
14011 
14067 
14124 
14180 



14237 
14293 
14350 
14406 
14462 



14519 
14575 
14631 
14688 
14744 



14800 
14856 
14913 
14969 
15025 



15081 
15137 
15193 
15249 
15305 



15361 
15417 
15473 
15529 
15585 



9 15641 
Log.Exs, 



09823 
09872 
09920 
09969 
10018 
10067 
10115 
10164 
10213 
10261 



10310 
10358 
10407 
10455 
10504 



10552 
10601 
10649 
10697 
10746 



10794 
10842 
10890 
10939 
10987 




11754 
11801 
11849 
11897 
11944 



11992 
12039 
12087 
12134 
12182 



12229 
12277 
12324 
12371 
12419 



12466 
12513 
12560 
12608 
12655 



9 12702 
Lg. Vers. 




16198 
16254 
16309 
16365 
16420 



16476 
16531 
16587 
16642 
16698 



16753 
16808 
16864 
16919 
16974 



17029 
17085 
17140 
17195 
17250 



17305 
17361 
17416 
17471 
17526 
17581 
17636 
17691 
17746 
17801 




18403 
18458 
18513 
18567 
18622 



18676 
18731 
18786 
18840 
18894 



9 18949 



Log.Exs. 



P. P. 





57 


57 


5€ 


6 


5.7 


5-7 


5 


7 


6 


7 


6 


6 


6 


8 


7 


6 


7 


6 


7 


9 


8 


6 


8 


5 


8. 


10 


9 


6 


9 


5 


9 


20 


19 


1 


19 





18 


30 


28 


7 


28 


5 


28. 


40 


38 


3 


38 





37- 


50 


47 


9 


47 


5 


47. 



55_ 



5 


5 


5 


6 


5 


6. 


7 


4 


7 


8 


3 


8 


9 


2 


9. 


18 


5 


18 


27 


7 


27 


37 





36 


46 


2 


45 



55 

5 

• 4 
•3 
•2 

• 1 
3 





54 


54 


6 


5 4 


5-4 


7 


6 


3 


6 


3 


8 


7 


2 


7 


2 


9 


8 


2 


8 


1 


10 


9 


1 


9 





20 


18 


1 


18 





30 


27 


2 


27 





40 


36 


3 


36 





50 


45 


4 


45 








51 


50 


5C 


6 


5-1 


5.0 


5 


7 


5 


9 


5 


9 


5 


8 


6 


8 


6 


7 


6 


9 


7 


6 


7 


6 


7 


10 


8 


5 


8 


4 


8 


20 


17 





16 


8 


16 


30 


25 


5 


25 


2 


25 


40 


34 





33 


5 


33 


50 


42 


5 


42 


1 


41 



49_ 

4-9 



16 
24 
33 
41 

48 

4 

5 

6 

7 

8 
16 
24 
32 
40 



49 

4-9 
7 
5 
3 
1 
3 
5 
6 



7 

8 
16 
24 
32 
40 

47 

4 
5 
6 
7 
7 

15 

23 

31 

39. 

P.P. 



48 
48 



47 



•7 


4 


7 


• 5 


5 


5 


• 3 


6 




•1 


7 





.9 


7 


8 


• 8 


15 


6 


• 7 


23 


5 


• 6 


31 


3 


■ 6 


39 


1 



652 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS* 
30° 31° 



Lg. Vers. D Log.Exs. D Lg. Vers. D Log.Exs. z> 



12702 
12749 
12796 
12843 
12890 



12937 
12984 
13031 
13078 
13125 



13172 
13219 
13266 
13313 
13359 



13406 
13453 
13500 
13546 
13593 



13639 
13686 
13733 
13779 
13826 
13872 
13919 
13965 
14011 
14058 



14104 
14151 
14197 
14243 
14289 



14336 
14382 
14428 

l<"=t71 
11-520 



14566 
14612 
14658 
14704 
14750 



14796 
14842 
14888 
14934 
14980 
15026 
15071 
15117 
15163 
15209 



15254 
15300 
15346 
15391 
15437 



9-15483 
' Lg. Vers. 



1894? 
•19003 
•19058 
•19112 
•19167 




.19763 
.19817 
.19871 
.19925 
.19979 



•20033 
.20087 
.20141 
.20195 
.20249 



.20303 
.20357 
.20411 
.20465 
•20518 



•20572 
•20626 
•20680 
•20733 
•20787 



•20841 
•20894 
•20948 
•21002 
•21055 



•21109 
•21162 
.21216 
•21269 
•21323 



•21376 
•21430 
•21483 
•21537 
•21590 



•21643 
•21697 
•21750 
•21803 
■21857 



•21910 
•21963 
•22016 
•22070 
•22123 



9-22176 
Log.Exs. 




15483 
15528 
15574 
15619 
15665 



15710 
15755 
15801 
15846 
15891 



15937 
15982 
16027 
16073 
16118 



16163 
16208 
16253 
16298 
16343 



16838 
16882 
16927 
16972 
17017 



17061 
17106 
17151 
17195 
17240 



17284 
1732? 
17373 
17418 
17462 




17950 
17994 
18038 
18082 
18126 



1817^) 



D LiTV^ 



22176 
2222? 
22282 
22335 
22388 



22441 
22494 
22547 
22600 
22653 



22706 
2275? 
22812 
22865 
22918 



22971 
23024 
23076 
23129 
23182 



23235 
23287 
23340 
23393 
23446 



23498 
23551 
23603 
23656 
23709 



23761 
23814 
23866 
2391? 
23971 



24024 
24076 
24128 
24181 
24233 



24285 
24138 
24390 
24442 
24495 



24547 
2459? 
24651 
24704 
24756 



24808 
24860 
24912 
24964 
25016 



25068 
2512^ 
25172 
25224 
25276 



25328 
Log.Exs. 



O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

li 

15 

16 

17 

18 

_19 

30 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39_ 

40 

41 

42 

43 

li 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59_ 

60 



P.P. 





55 


54 


6 


5^4 


5.4 


7 


6 


3 


6.3 


8 


7 


2 


7.2 


9 


8 


2 


8.1 


10 


9 


1 


9.0 


20 


18 


1 


18.0 


30 


27 


2 


27.0 


40 


36 


3 


36.0 


50 


45 


4 


45.0 



53_ 

5.3 

6.2 

7.1 

8.0 

8.9 

17.8 

26.7 

35.6 

44^6 





53 


53 


5- 


3 


6 


5^3 


5.2 


5.2 


7 


6 


2 


6-1 


6 





8 


7 





7.0 


6 


9 


9 


7 


9 


7.9 


7 


8 


10 


8 


8 


8-7 


8 


6 


20 


17 


6 


17^5 


17 


3 


30 


26 


5 


26-2 


26 





40 


35 


3 


35^0 


34 


6 


50 


44 


1 


43^7 


43 


3 





47 


47 


6 


4.7 


4.71 


7 


5 


5 


5 


5 


8 


6 


3 


6 


2 


9 


7 


1 


7 





10 


7 


9 


7 


8 


20 


15 


8 


15 


6 


30 


23 


7 


23 


5 


40 


31 


5 


31 


3 


50 


39 


6 


39 


1 



7 
8 
9 
10 
20 
30 
40 
50 



46 

4-6 
3 
I 
6 
7 
15 
23 
30 
38 



45_ 

4^5 



46_ 

4-6 

5.4 

6.2 

7.0 

7.7 

15^5 

23.2 

31.0 

.7 



45 

4-5 



10 
20 
30 
40 
50 



44, 


4^4| 


5 


2 


5 


9 


6 


7 


7 


4 


14 


8 


22 


2 


29 


6 


37 


1 



44 

4.4 

5^1 

5-8 

6.6 

7-3 

14-6 

22.0 

29.3 

36.6 



P.P. 



653 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
32° 33° 




P. P. 





53 


51 


6 


5.2 


5.11 


7 


6 





6 





8 


6 


9 


6 


8 


9 


7 


8 


7 


7 


10 


8 


6 


8 


6 


20 


17 


3 


17 


1 


30 


26 





25 


7 


40 


34 


6 


34 


3 


50 


43 


3 


42 


9 



51 

5.1 

5.9 

68 

7-6 

8-5 

17.0 

25.5 

34-0 

42.5 





50 


50 


49 


6 


5.0 


5.0 


4.9 


7 


5 


9 


5 


8 


5 


8 


8 


6 


7 


6 


6 


6 


6 


9 


7 


6 


7 


5 


7 


4 


10 


8 


4 


8 


3 


8 


2 


20 


16 


8 


16 


6 


16 


5 


30 


25 


2 


25 





24 


7 


40 


33 


6 


33 


3 


33 




50 


42 


1 


41 


6 


41 


2 



44 

4-4 



43_ 

4-3 



43 

4.3 



42 


/ 


2 


4 


I 


4.2 


4 


2 


4.1 


4 


9 


4 


9 


4 


8 


5 


g 


5 


6 


5 


5 


6 


4 


6 


3 


6 


2 


7 


1 


7 





6 


9 


14 


1 


14 





13 


8 


21 


2 


21 





20 


7 


28 


3 


28 





27 


6 


35 


4 


35 





34 


6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



41 

4.1 
48 

5.4 
6.1 
6.8 
13.6 
20.5 
27-3 
34.1 



P. P. 



654 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
34° 35° 



Lg.Vers. D Log.Exs. 1> Lg. Vers. D Log.Exs. D 



23290 
23331, 
23372 
23414 
23455 



23496 
23537 
23579 
23620 
23661 



23702 
23743 
23784 
23825 
23866 



23907 
23948 
23989 
24030 
24071 



24112 
24153 
24194 
24235 
24275 



24316 
24357 
24398 
24438 
24479 



24520 
24561 
24601 
24642 
24682 



24723 
24764 
24804 
24845 
24885 



24926 
24966 
25007 
25047 
25087 



25128 
25168 
25209 
25249 
25289 

25329 
25370 
25410 
25450 
25490 



25531 
25571 
25611 
25651 
25691 



25731 



' Lg. Vers. 



9-31432 
31482 
31532 
31582 
31632 



31681 
31731 
31781 
31831 
31880 



31930 
31980 
32029 
32079 
32129 

32178 
32228 
32277 
32327 
32377 



32426 
32476 
32525 
32575 
32624 



32673 
32723 
32772 
32822 
32871 



32920 
32970 
33019 
33069 
33118 



33167 
33216 
33266 
33315 
33364 



33413 
33463 
33512 
33561 
33610 



33659 
33708 
33758 
33807 
33856 



33905 
33954 
34003 
34052 
34101 



34150 
34199 
34248 
34297 
34346 



34395 
Log.Exs. 



5 9. 



•28099 



l> Lg.Vers, 



34395 
34444 
34492 
34541 
34590 



34639 
34688 
34737 
34785 
34834 



34883 
34932 
34980 
34029 
35078 



35127 
35175 
35224 
35273 
35321 



35370 
35419 
35467 
35516 
35564 



35613 
35661 
35710 
35758 
35807 



35855 
35904 
35952 
36001 
36049 



36098 
36146 
36194 
36243 
36291 



36340 
36388 
36436 
36484 
36533 



36581 
36629 
36678 
36726 
36774 




Log.Exs. I> 



O 

1 

2 
3 

4 

5 

6 

7 

8 

_9^ 

10 

11 

12 

13 

JA 

15 

16 

17 

18 

19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
2^ 

30 

31 
32 
33 
-34 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45 
46 
47 
48 
ii 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59^ 
60 



P.P. 



50 

5.0| 

5 

6 

7 

8 
16 
25 



41 



49_ 

4-9 

5 

6 

7 

8 
16 
24 
33 
41 



49 

4.9 



48 



4 


8 


4. 


5 


6 


5. 


6 


4 


6. 


7 


3 


7. 


8 


1 


8 


16 


1 


16 


24 




24 


32 


3 


32 


40 


4 


40 



41_ 

41 





40 


40 


6 


4.0 


40 


7 


4 


7 


4 


6 


8 


5 


4 


5 


3 


9 


6 


1 


6 





10 


6 


7 


6 


6 


20 


13 


5 


13 


3 


30 


20 


2 


20 





40 


27 





26 


6 


50 


33 


7 


33 


3 



6 

7 

8 

9 

10 

20 

30 

40 

50 



39 

3-9 



39 

3 

4 



38 

38 



6 
7 
8 
9 

10 
20112 
30 19 
40 25 
50132 
P. P. 



48 
8 
6 
4 
2 






41 

41 



655 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
36° 37° 



Lg. Vers. ^> Log.Exs. I> Lg.Vers, I> Log.Exs. -Z> 



O 

1 
2 
3 

j4 
5 
6 
7 
8 

_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



•28099 
.28138 
.28177 
.28816 
■28255 
•28293 
•28332 
•28371 
•28410 
^8448 
28487 
•28526 
•28564 
•28603 
.28642 



.28680 
•28719 
•28757 
•28796 
28835 



28873 
.28912 
•28950 
•28988 
•29027 



29065 
•29104 
•29142 
•29180 
^921_9 
929257 
•29295 
.29334 
.29372 
•29410 



9 29448 

.29487 

•29525 

.29563 

29601 



9 .29639 
29677 
29715 
29754 
29792 



9-29830 
.29868 
.29906 
•29944 
•29982 



9^30020 
.30057 
.30095 
.30133 
.30171 




iLg. Vers 




38023 
38071 
38119 
38166 
38214 



38262 
38310 
38357 
38405 
38453 



38501 
38548 
38596 
38644 
38692 
38739 
38787 
38834 
38882 
J893P 
38977 
39025 
39072 
39120 
39168 



39215 
39263 
39310 
39358 
39405 

39453 
39500 
39548 
39595 
39642 



39690 
39737 
39785 
3983? 
39879 



39927 
39974 
40021 
40069 
40116 



9.40163 
Log.Exs. 



9.30398 
.30436 
.30474 
•30511 
• 30^49 

9^30587 
•30624 
•30662 
•30700 
.30737 



9.30775 
•30812 
•30850 
•30887 
•30925 

9^30962 
•31000 
•31037 
•31075 
■31112 



9. 31150 

•31187 

•31224 

31262 

312^9 

9 31336 

•31374 

•31411 

•31448 

■31485 

931523 

31560 

31597 

31634 

31671 

9 •31708 

•31746 

.31783 

•31820 

31857 



931894 
•31931 
.31968 
.32005 
■ 3204 2 

9 . 32079 
.32116 
.32153 
.32190 
.3222 7 

9 . 32263 
.32300 
.32337 
.32374 
.32411 



9 . 32447 
.32484 
.32521 
.32558 
.32594 



9-32631 



Lg. Vers. 



37 
38 
37 
37 
38 
37 
37 
38 
37 
37 
37 
37 
37 
37 

37 
37 
3Z 
37 
37 
37 
37 
37 
37 
37 

37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
36 
37 
37 
36 
37 
36 
37 
36 
37 
36 
37 




.40399 
.40447 
. 40494 
•40541 
40588 



.40635 
.40682 
•40730 
.40777 
■40824 



■40871 
•40918 
•40965 
•41012 
43C59 




9. 41810 
•41857 
•41904 
•41951 
^1998 



9^42044 
.42091 
.42138 
.42185 
.42231 



1.42278 
.42325 
.42372 
.42418 
.42465 



•42512 
.42558 
.42605 
•42652 
•42698 



9.42745 
.42792 
.42838 
.42885 
.42931 



9^42978 



47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

47 

46 

47 

47 

47 

46 

47 

47 

46 

47 

47 

46 

47 

46 

47 

46 

47 

46 

47 

46 

47 

46 

46 

47 

46 

46 

46 

47 

46 

46 

46 

46 



J> Log.Exs. 
656 



D 



P.P. 





1 

2 

3 
__4 

5 

6 

7 

8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
il 
20 
21 
22 
23 
_24 
25 
26 
27 
28 
_29 
30 
31 
32 

M 

35 
36 
37 
38 
39. 
40 
41 
42 
43 
M 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



48_ 



10 
20 
30 
40 
50 



4 


8 


4^ 


5 


6 


5- 


6 


4 


6. 


7 


3 


7- 


8 


1 


8. 


16 


1 


16. 


24 


2 


24. 


32 


3 


32. 


40 


4 


40. 



47 



4.71 


5 


5 


6 


3 


7 


1 


7 


9 


15 


8 


23 


7 


31 


6 


39 


6 



7 
8 
9 
10 
20 
30 
40 
50 



48 
8 
6 
4 
2 






47 

4.7 

5.5 

6.2 

7.0 

7.8 

15-6 

23.5 

31.1 

39.1 

46 

4.6 

5.4 

6.2 

7.0 

7.7 
15.5 
23.2 
31.0 
38-7 



39 

3 

4 

5 

5 

6 
13 
10 
26 
32 

38 



38 

3.8 



3 


8 


4 


4 


5 





5 


7 


6 


3 


12 


6 


19 





25 


3 


31 


6 



37 



3 


7 


4 


3 


4 


9 


5 


5 


6 


1 


12 


3 


18 


5 


24 


6 


30 


8 



37. 

3.7 

4.4 

5.0 

5-6 

6.2 

12.5 

18.7 

25.0 

31-2 

36 

3-6 

4-2 

4.8 

5.5 

G-l 

12.1 

18-2 

24.3 

30.4 



P. P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
38° 39° 



Lg. Vers 



932631 
32668 
32704 
32741 
32778 



32814 
32851 
32888 
32924 
32961 



32997 
33034 
33070 
33107 
33143 



33180 
33216 
33252 
33289 
33325 



33361 
33398 
33434 

33470 
33507 



33543 
33579 
33615 
33652 
33688 



33724 
33760 
33796 
33833 
33869 

33905 
33941 
33977 
34013 
34049 



34085 
34121 
34157 
34193 
34229 



34265 
34301 
34337 
34373 
34408 



34444 
34480 
34516 
34552 
34587 



34623 
34659 
34695 
34730 
34766 



34802 



' Lg. Vera 



2> 



Log.Exs. 



42978 
43024 
43071 
43118 
43164 



43211 
43257 
43304 
43350 
43396 



43443 
43489 
43536 
43582 
4 3629 

43675 
43721 
43768 
43814 
43861 



43907 
43953 
43999 
44046 
44092 



44138 
44185 
44231 
44277 
44323 



44370 
44416 
44462 
44508 
44554 



44601 
44647 
44693 
44739 
44785 




45292 
45338 
45384 
45430 
45476 



45522 
45568 
45614 
45660 
45706 



9. 45752 



jy Log.Exs, 



Lg. Vers 



34802 
34837 
34873 
34909 
34944 
34980 
35016 
35051 
35087 
35122 



35158 
35193 
35229 
35264 
35300 



35335 
35370 
35406 
35441 
35477 



36040 
36076 
36111 
36146 
36181 



36216 
36251 
36286 
36321 
36356 



36391 
36426 
36461 
36495 
36530 
36565 
36600 
36635 
36670 
36705 




JD 



-O iLg. Vers.) i> Log.Exs, 



Log.Exs, 



45752 
45797 
45843 
45889 
4593 5 

45981 
46027 
46073 
46118 
46164 



46210 
46256 
46302 
46347 
46393 



46439 
46485 
46530 
46576 
46622 



46668 
46713 
46759 
46805 
46850 
46896 
46942 
46987 
47033 
47078 
47124 
47170 
47215 
47261 
47306 



47352 
47398 
47443 
47489 
47534 



47580 
47625 
47671 
47716 
47762 



47807 

47852 

47898 

4794: 

47989 



48034 
48080 
48125 
48170 
48216 



48261 
48306 
48352 
48397 
48442 



48488 



^1 





1 
2 
3 
_4 
5 
6 
7 
8 
9 

10 

11 
12 
13 
il 
15 
16 
17 
18 
19 

20 

21 
22 
23 
_2i 
25 
26 
27 
28 
-29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39, 
40 
41 
42 
43 
44 

45 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59_ 

60 



P.P. 



47 



4 


7 


5 


5 


6 


2 


7 





7 


3 


15 


6 


23 


5 


31 


3 


39 


1 



46_ 

4-6 

5.4 

6-2 

7.0 

7.7 

15.5 

23.2 

31.0 

38.7 



5 
6 
6 
7 
15 
30i23 
4030 
50l38 



46 

4.6 



45 

4.5 

5.3 

6 

6 8 

7.6 

15.1 

_ 22.7 

6|30.3 

3I37.9 



6 


4. 


7 


5. 


8 


6- 


9 


6 


10 


7. 


20 


15- 


30 


22 


40 


30 


50 


37. 



e 
7 

8 
9 

10 
20 
30 
40 
50 



37 


3-71 


4 


3 


4 




5 


5 


6 


1 


12 


3 


18 


5 


24 


g 


30 


8 



36 


3 


6 


4 


2 


4 


8 


5 


4 


6 





12 





18 





24 





30 






35 


3 


5 


4 


1 


4 


6 


5 


2 


5 


8 


11 


6 


17 


5 


23 


3 


29 


1 



45 

5 
2 

7 
5 

5 

5 

36 

3.6 

4.2 

4.8 

5.5 

6.1 

12.1 

18.2 

24.3 

30.4 

35 

3.5 

4.1 

4.7 

5.3 

5.9 

11.8 

17.7 

23.6 

29.6 

35 

3.4 

4.0 

4.6 

5.2 

5.7 

11.5 

17.2 

23.0 

28. 7 



P.P. 



Go7 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
40° 41° 



O 

1 
2 
3 

_4 
5 
6 
7 
8 

_9 

10 

11 
12 
13 
14 

15^ 
16 
17 
18 
19 



20 

21 
22 
23 
24 
25 
26 
27 
28 
2i 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



40 

41 
42 
43 
44 
45 
46 
47 
48 
49 



50 

51 
52 
53 
54 
55 
56 
57 
58 
5i 

60 



Lg. Vers, 




36913 
36948 
36982 
37017 
37052 



37604 
37639 
37673 
37707 
3774^ 

37776 
37810 
37845 
37879 
3791| 
37947 
37982 
38016 
38050 
38084 



38118 
38153 
38187 
38221 
38255 



38289 
38323 
38357 
38391 
38425 

38459 
38493 
38527 
38561 
38595 



38629 
38663 
38697 
38731 
38765 



38799 
38833 
38866 
38900 
38934 



9-38968 



Lg.Ve 



J> 



Log.Exs. 



9. 48488 
48533 
48578 
48624 
48669 




49392 
49437 
49482 
49527 
49572 



49618 
49663 
49708 
49753 
49798 
49843 
49888 
49933 
49978 
_50p^ 
50068 
50113 
50158 
50203 
50248 



50293 
50338 
50383 
50427 
50412 

50517 
50562 
50607 
50652 
50697 
50742 
50787 
50831 
50876 
50921 



50966 
51011 
51055 
51100 
51145 



9 51190 
Log.Exs, 



D 



Lg. Vers, 



9.38968 
39002 
39035 
39069 
3_9103 
39137 
39170 
39204 
39238 
39271 
39305 
39339 
39372 
39406 
39439 
39473 
39507 
39540 
39574 
39607 



39641 
39674 
39708 
39741 
39774 



39808 
39841 
39875 
39908 
39941 

39975 
40008 
40041 
4C075 
40108 



40141 
40175 
40208 
40241 
40274 



40307 
40341 
40374 
40407 
40440 



40473 
40506 
40540 
40573 
40606 



40639 
40672 
40705 
40738 
40771 



40804 
40837 
40870 
40903 
40936 



9-40969 



Lg. Vers, 



Log.Exs, 



51190 
51235 
51279 
51324 
51369 



51414 
51458 
51503 
51548 
51592 



51637 
51682 
51726 
51771 
51816 
51860 
51905 
51950 
51994 
52039 



52084 
52128 
52173 
52217 
52262 
52306 
52351 
52396 
52440 
52485 



52529 
52574 
52618 
52663 
52707 



52752 
52796 
52841 
52885 
52930 



52974 

53018 

53063 

5310 

53152 



53196 
53240 
53285 
53329 
53374 



53418 

5346 

53507 

53551 

53595 



53640 
53684 
53728 
53773 
53817 



53861 



1> Log.Exs, 

G58 



2 
3 
_4 

5 

6 

7 

8 

_9_ 
10 
11 
12 
13 
_14 
15 
16 
17 
18 
19 

20 

21 
12 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 
42 
43 
4_i 
45 
46 
47 
48 
49 

50 
51 
52 
53 

_54 
55 
56 
57 
58 

_59 

60 



D 



P.P. 



45_ 

45 



45 

4-5 
2 

7 
5 

5 

5 



44_ 44 

.4 
1 
8 
6 

•3 
6 

•Q 
3 
6 



4.4 


4. 


5 


2 


5. 


5 


9 


5. 


6 


7 


6. 


7 


4 


7. 


14 


3 


14. 


22 


2 


22. 


29 


6 


29. 


37 


1 


36. 



35 

3.5 



34 



3 


4 


3 


9 


4 


5 


5 


1 


5 


6 


11 


3 


17 





22 


6 


28 


3 



31 

3-4 

4.0 

4.6 

5.2 

5.7 

11.5 

17.2 

23.0 

28-7 



33 

3.3 
3.9 

4-4 
5.0 
5-6 
11.1 
16-7 
22.3 
27.9 



33 

3.3 
3.8 
4.4 
4.9 

5.5 
20:11.0 
30 16.5 
4022.0 
50127.5 



P. P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
43° 43° 



Lg. Vers 



2> 



Log.Exs 



I) 



Lg. Vers 



l> iLog.Exs 



P.P. 



40969 
41001 
41034 
41067 
41100 



41133 
41166 
41199 
41231 
41264 



41297 
41330 
41362 
41395 
41428 



41461 
41493 
41526 
41559 
41591 



41624 
41657 
41689 
41722 
41754 



41787 
41819 
41852 
41885 
41917 



41950 
41982 
42014 
42047 
42079 



42112 
42144 
42177 
42209 
42241 



42274 
42306 
42338 
42371 
42403 



42435 
42467 
42500 
42532 
42564 



42596 
42629 
42661 
42693 
42725 



42757 
42789 
42822 
42854 
42886 



42918 
. Vers. 



53861 
53906 
53950 
53994 
54038 



54083 
54127 
54171 
54215 
54259 



54304 
54348 
54392 
54436 
54480 



54525 
54569 
54613 
54657 
54701 



54745 
54790 
54834 
54878 
54922 



54966 
55010 
55054 
55098 
55142 



55186 
55230 
55275 
55319 
55363 



55407 
55451 
55495 
55539 
55583 



55627 
55671 
55715 
55759 
55803 



55847 
55890 
55934 
55978 
56022 



56066 
56110 
56154 
56198 
56242 



56286 
56330 
56374 
56417 
56461 



9-56505 
Log.Exs. 



42918 
42950 
42982 
43014 
43046 



43078 
43110 
43142 
43174 
43206 



43238 
43270 
43302 
43334 
43365 



43397 
43429 
43461 
43493 
43525 



43557 
43588 
43620 
43652 
43684 



43715 
43747 
43779 
43810 
43842 



43874 
43906 
43937 
43969 
44000 



44032 
44064 
44095 
44127 
44158 



44190 
44221 
44253 
44284 
44316 



44347 
44379 
44410 
44442 
44473 



44504 
44536 
44567 
44599 
44630 



44661 
44693 
44724 
44755 
44787 



44818 
Lg. Vers. 



9-56505 
56549 
56593 
56637 
56680 



56724 
56ro8 
56812 
56856 
56899 



56943 
56987 
57031 
57075 
57118 



57162 
57206 
57250 
57293 
57337 



57381 
57424 
57468 
57512 
57556 



57599 
57643 
57687 
57730 
57774 



57818 
57861 
57905 
57949 
57992 



58036 
58079 
58123 
58167 
58210 



58254 
58297 
58341 
58385 
58428 



58472 
58515 
58559 
58602 
58646 



58689 
58733 
58776 
58820 
58864 



58907 
58951 
58994 
59037 
59081 



9 59124 



Log.Exs, 



41 



44 

4 
1 
8 
6 
3 
6 

3 



43 43 



4.4 


4. 


5.2 


5. 


5.9 


5. 


6.7 


6. 


7.4 


7. 


14-8:14- 


22-2 22. 


29.6,29. 


37.1I36. 



4-3 


4 


5 


1 


5 


5 


8 


5 


6 


5 


6 


7 


2 


7 


14 


5 


14 


21 


7 


21. 


29 





28. 


36 


2 


35. 



33 

3-3 



33 

3.2 
38 

4-3 
4.9 
5.4 
10-8 
5,16-2 
21.6 
5I27.I 



32 



3 


2 


3 


3 


7 


3 


4 


2 


4 


4 


8 


4 


5 


3 


5 


10 


6 


10 


16 





15 


21 




21- 


26 


6 


26- 



31_ 

1 
7 
2 
7 
2 
5 
7 

2 



6 


3 


7 


3 


8 


4 


9 


4 


10 


5. 


20 


10 


30 


15 


40 


20- 


50 


25. 



31 

1 
6 
1 
6 
1 
3 
5 



P. P. 



659 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
44° 45° 



Lg.Vers. 



44818 
44849 
44880 
44912 
44943 



44974 
45005 
45036 
45068 
45099 



45130 
45161 
45192 
45223 
45254 



45285 
45316 
45348 
45379 
45410 



45441 
45472 
45503 
45534 
45565 




45904 
45935 
45966 
45997 
46027 



46058 
46089 
46120 
46150 
46181 



46212 
46242 
46273 
46304 
46334 



46365 
46396 
46426 
46457 
46487 



n 



46518 
46549 
46579 
46610 
46640 
9-46671 
Lg. Vers. 



9-59124 
59168 
59211 
59255 
59298 



Log.Exs, 



59342 
59385 
59429 
59472 
59515 



59559 
59602 
59646 
59689 
59732 



59776 
59819 
59863 
59906 
59949 



59993 
60036 
60079 
60123 
60116 
60209 
60253 
60296 
60339 
60313 
60426 
60469 
60512 
60556 
60599 
60642 
60685 
60729 
60772 
60815 




2> 



Lg.Vers, 



61722 
Log.Exs, 



46671 
46701 
46732 
46762 
4 6793 
46823 
46853 
46884 
46914 
46945 



46975 
47005 
47036 
47066 
47096 



47127 
47157 
47187 
47218 
47248 



47278 
47308 
47339 
47369 
47399 



47429 
47459 
47490 
47520 
47550 



47580 
47610 
47640 
47670 
47700 



47731 
47761 
47791 
47821 
^7851 



47881 
47911 
47941 
47971 
48001 




48329 
48359 
48389 
48419 
48449 
48478 



Z> Lg.Vers, 



n 



Log.Exs, 



61722 
61765 
61808 
61852 
61895 



61938 
61981 
62024 
62067 
62110 



62153 
62196 
62239 
62282 
62326 



62369 
62412 
62455 
62498 
62541 



62584 
62627 
62670 
62713 
62756 



2> 




63229 
63272 
63315 
63358 
63401 



63443 
63486 
63529 
63572 
63615 



63658 
63701 
63744 
63787 
63830 



63873 
63915 
63958 
64001 
64044 




D jLog.Exs. 
660 



o 

1 

2 
3 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
11 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
2^ 

30 

31 
32 
33 
_3i 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45 
46 
47 
48 

50 

51 
52 
53 
54 
55 
56 
57 
58 
-59 
60 



P. P. 



43_ 

4.3 

5-1 

5-8 

6 

7 

14 

21 

29 

36 



43 

4-3 



43_ 

4-2 



6 

7 

8 

9 

10 

20 

30 

40 



3 

4 

4 

5 

10 

15 

21 

50 26 



31 

3l 



31 

3.1 





30 


30 


6 


3-0 


3-0 


7 


3 


5 


3 


5 


8 


4 





4 





9 


4 


6 


4 


5 


10 


5 


1 


5 





20 


10 


1 


10 





30 


15 


2 


15 





40 


20 


3 


20 





50 


25 


4 


25 






29 

9 
4 
9 
4 
9 
8 
7 
6 



P. P. 



6 


2. 


7 


3. 


8 


3. 


9 


4. 


10 


4. 


20 


9. 


30 


14. 


40 


19. 


50 


24. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

46° 47° 




Log.Exs 



66864 
66907 
66950 
66992 
67035 



67077 
67120 
67162 
67205 
67248 



67290 
67333 
67375 
67418 
67460 



67503 
67546 
67588 
67631 
67673 



67716 
67758 
67801 
67843 
67886 
67928 
67971 
68013 
68056 
68098 



68141 
68183 
68226 
68268 
68311 



68353 
68396 
68438 
68481 
68523 



68566 
68608 
68651 
68693 
68735 



68778 
68820 
68863 
68905 
6.894^ 
68990 
69033 
69075 
69117 
6 9160 
69202 
69245 
69287 
69330 
69372 
9-69414 
Log.Exs. 





1 

2 
3 
4 
5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
11 
35 
36 
37 
38 
li 
40 
41 
42 
43 
44 

45 
46 
47 
48 
ii 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59_ 
60 



P. P. 



43 



7 
8 
9 
10 
20 
30 
40 
50 



4-3 


4 


5.0 


4 


5.7 


5 


6-4 


6 


7.1 


7. 


14.3 


14 


21.5 


21. 


28.6 


28. 


35.8 


35. 



42 

2 
9 
6 
4 
1 
1 
2 
3 
4 



6 


4 


7 


4 


8 


5. 


9 


6. 


10 


7. 


20 


14. 


30 


21. 


40 


28. 


50 


35. 



43 

2 
9 
6 
3 








30 



9 
10 
20 
30 
40 
50 



3 





2 


3 


5 


3 


4 





3 


4 


5 


4 


5 





4 


10 





9- 


15 





14. 


20 





19. 


25 





24. 



10 
20 
30 14 
40 19 
50 24 



29 

2.9 

3 

3 

4 

4 

9 



29 

9 
4 
9 
4 
9 
8 
7 



28 



28 
28 



P. P. 



661 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
48° 49° 




662 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
< 60° 51° 



Lg.Vers. D Log.Exs. /> Lg.Vers. 2> Log.Exs l> 



9.55698 
.55725 

• 55751 
.55778 

• 55805 



9-56234 
•56261 
.56288 
•56315 
•56311 



•55832 

55859 

55886 

•55913 

.55940 



•55906 
•55993 
•56020 
•56047 
•56074 



•56101 
.56127 
.56154 
.56181 
.56208 



9. 56368 
.56395 
.56421 
. 56448 
•56475 



9 •56501 
•56528 
•56554 
.56581 
•56608 



9-56634 
•56661 
•56687 
•56714 
•56741 



9-56767 
•56794 
•56820 
•56847 
•56873 
9-56900 
Lg. Vers, 



1.74486 
.74528 
•74570 
.74612 
-74654 



1-74696 
•74739 
•74781 
•74823 
•74865 



.56900 
.56926 
.56953 
.56979 
.57005 



9.74907 
• 74949 
•74991 
•75033 
•75076 



9.75118 
•75160 
•75202 
•75244 
.75286 



9.75328 
.75370 
.75413 
.75455 
•75497 



9^75539 
.75581 
.75623 
.75665 
.75707 



9.75750 
.75792 
.75834 
.75876 
•75918 



9-75960 
.76002 
.76044 
•76086 
.76128 



9.76171 
.76213 
.76255 
.76297 
.76339 



9.76381 
.76423 
.76465 
.76507 
.7654S 



9.76592 
•76634 
•76676 
•76718 
.76760 



.76802 
.76344 
.768«»6 
.76928 
•76970 



9-77012 

J> Log.Exs 



1.57032 
•57058 
•57085 
•57111 
.57138 



9.57164 
•57190 
•57217 
•57243 
•57269 



9.57690 
.57716 
.57742 
.57768 
•57794 



9-57296 
57322 
57348 
57375 
57401 



9^57427 
.57454 
.57480 
.57506 
•57532 



9^57559 
•57585 
.57611 
.57637 
.57664 



9.57821 
.57847 
.57873 
.57899 

•57925 



9.57951 
.57977 
•58003 
•58029 
.58055 



9.58082 
58108 
58134 
58160 
58186 



9.58212 
.58238 
.58264 
•58290 
.58316 



9-58342 
.58367 
-58393 
.58419 
•58445 



9-58471 
D Lg. Vers. 



9-77012 
.77055 
-77097 
.77139 
•77181 



9-77223 
.77265 
•77307 
•77349 
•77391 



9.78275 
.78317 
.78359 
.78401 
•78443 

9 •78485 
.78527 
.78569 
•78611 
• 7865!^ 



9.78064 
.78107 
.78149 
.78191 
•78233 



•78696 
•78738 
•78780 
•78822 
.78864 



9.78906 
.78948 
.78990 
.79032 
.79074 



9.79116 
.79158 
.79200 
.79242 
.79285 



•79327 
•79369 
•79411 
•79453 
-79495 



9-79537 
l> Log.Exs. 

~663 " 



P.P. 



6 
7 
8 

9 
10 
20 
30 
40 
50 



10 
20 
30 
40 
50 



4^_ 

4.2 
4.9 
5.6 
6.4 

7.1 
14.1 
21.2 
28-3 
35-4 



27 

2-7 



43 

4.2 

4^9 

5^Q 

6.3 

7.0 

14-0 

21.0 

28.0 

35. 



27 





26 


26 


6 


2.6 


2.6 


7 


3 


1 


3 


6 


8 


3 


5 


3 


4 


9 


4 





3 




10 


4 


4 


4 


3 


20 


8 


8 


8 


6 


30 


13 


2 


13 




40 


17 


6 


17 


3 


50 


22 


1 


21 


6 





25 


6 


2.^ 


7 


3.0 


8 


3.4 


9 


3-8 


10 


4-2 


20 


8-5 


30 


12-7 


40 


17.0 


50 


21-2 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 
53° 53° 



10 

11 
12 
13 
14 
15 
16 
17 
18 
19 



20 

21 
22 
23 
24 
25 
26 
27 
28 
29. 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 
51 
52 
53 
5i 
55 
56 
57 
58 
59 
60 



Lg. Vers 



58471 
58497 
58523 
58549 
58575 



58601 
58626 
58652 
58678 
58704 



58''30 
58755 
58781 
58807 
58833 



58859 
58884 
58910 
58936 
58962 



58987 
59013 
59039 
59064 
59090 



59116 
59141 
59167 
59193 
59218 



59244 
59270 
59295 
59321 
59346 



59372 
59397 
59423 
59449 
59474 




59881 
59907 
59932 
59958 
59983 



60008 



' Lg. Vers 



D 



Log.Exs. 




79958 
80000 
80042 
80084 
8 0126 
80168 
80210 
80252 
80294 
80S36 



80378 
80420 
80463 
80505 
80547 



80589 
80631 
80673 
80715 
80757 



80799 
80841 
80883 
80925 
80968 



81010 
81052 
81094 
81136 
81178 



81220 
81262 
81304 
81346 
81388 



81430 
81473 
81515 
81557 
81599 



81641 
81683 
81725 
81767 
81809 




Log.Exs. 



n 



Lg.Vers. 



60008 
60034 
60059 
60084 
60110 



60135 
60160 
60185 
60211 
60236 



60261 
60286 
60312 
60337 
60362 



60387 
60412 
60438 
60463 
60488 



60513 
60538 
60563 
60589 
6C614 



60639 
60664 
60689 
60714 
60739 



60764 
60789 
60814 
60839 
6C864 



60889 
60914 
60939 
fC9ei 
er989 




61264 
61289 
61313 
61338 
61363 



61388 
61413 
61438 
61462 
61487 
961512 
Lg.Vers. 



Log.Exs. 



82062 
82104 
82146 
82188 
82230 



82272 
82315 
82357 
82399 
82441 



82483 
82525 
82567 
82609 
82651 



82694 
82736 
82778 
82820 
82862 



82904 
82946 
82988 
83031 
83073 



83536 
83578 
83620 
83663 
83705 



83747 
83789 
83831 
83873 
83916 



83958 
84000 
84042 
84084 
84.26 




Log.Exs 



n 



P.P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



42_ 



4 2 


4 


4 


9 


4. 


5 


6 


5. 


6 


4 


6. 


7 


1 


7. 


14 


1 


14. 


21 


2 


21. 


28 


3 


28. 


35 


4 


35. 



42 

2 
9 
6 
3 








26 



2 


5 


3 





3 


4 


3 


9 


4 


3 


8 


6 


13 





17 


3 


21 


6 



25^ 

2.5 

3.0 

3.4 

3 % 

4.2 

85 

12.7 

17.0 

21.2 



25 2?. 



7 
8 
9 
10 
20 
30 
40 
50 



2 


5 


2. 


2 


9 


2 


3 


3 


3. 


3 


7 


3. 


4 


1 


4 


8 


3 


8. 


12 


5 


12. 


16 


g 


16- 


20 


8 


20. 



P.P. 



GUI 



TABLE Vllf.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



54° 



55° 



Lg. Vers. 



n 



Log.Exsi 



z> 



Lg. Vers. 



Log.Exs. 



P.P. 



9.61512 
61537 
61562 
61586 
61611 



61636 
61661 
61685 
61710 
61735 



61760 
61784 
61809 
61834 
61858 



61883 
61908 
61932 
61957 
61982 



62006 
62031 
62055 
62080 
62105 



62129 
62154 
62178 
62203 
62227 



62252 
62276 
62301 
62325 
62350 



62374 
62399 
62423 
62448 
62472 



62497 
62521 
62546 
62570 
62594 



62619 
62643 
62668 
62692 
62716 



62741 
62765 
62789 
62814 
62838 




' Lg. Vers. 



84??0 
84632 
84675 
84717 
84759 



84801 
84843 
84886 
84928 
84970 



85012 
85054 
85097 
85139 
85181 



85223 
85265 
85308 
85350 
85392 



85434 
85476 
85519 
85561 
85603 



85645 
85688 
85730 
85772 
85814 



85857 
85899 
85941 
85983 
86026 



86068 
86110 
86152 
86195 
86237 



86279 
86321 
86364 
86406 
86448 




86913 
86956 
86998 
87040 
87082 



87125 



l^ Log.Exs, 



n 



62984 
63008 
63032 
63057 
6 3081 
63105 
63129 
63154 
63178 
63202 



63226 
63250 
63274 
63209 
63323 



63347 
63371 
63395 
63419 
63443 



63468 
63492 
63516 
63540 
63564 



63598 
63612 
63636 
63660 
63684 

63708 
63732 
63756 
63780 
63804 



63828 
63852 
63876 
63900 
63924 



63948 
63972 
63996 
64019 
64043 



64067 
64091 
64115 
64139 
64163 



64187 
64210 
64234 
64258 
64282 



64306 
64330 
64353 
64377 
6_4401 
64425 



Lg. Vers. 



.87125 
.87167 
.87209 
.87252 
.87294 



.87336 
.87379 
.87421 
.87463 
.87506 



.87548 
■87590 
.87633 
•87675 
.87717 



.87760 
.87802 
.87844 
.87887 
.87929 



•87971 
.88014 
.88056 
.88099 
.88141 



88183 
.88226 
.88268 
.88310 
•88353 



•88395 
.88438 
.88480 
.88522 
.88565 



•88607 
.88650 
•88692 
.88734 
•88777 



.88819 
.88862 
• 8890-^ 
•88947 



l> Log.Exs, 
665 



u 





1 

2 
3 
_4 
5 
6 
7 
8 
9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
11 
30 
21 
22 
23 
24 
25 
26 
27 
2E 
29, 
30 
31 
32 
33 
li 
35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
ii 
50 
51 
52 
53 
51 
55 
56 
57 
58 
59^ 
60 





4:2 


42 


6 


4.2 


4.2 


7 


4.9 


4.9 


8 


5.6 


5.6 


9 


6.4 


6.v3 


10 


7.1 


7.0 


20 


14.1 


14.0 


30 


21.2 


21.0 


40 


28.3 


28.0 


50 


35^4 


35.0 





25 


21 


6 


2.5 


2- 


7 


2.9 


2.1 


8 


3.3 


3.2 


9 


3.7 


3.! 


10 


4.1 


4.1 


20 


8.3 


30 


12.5 


12.2 


40 


16.6 


16. J^ 


50 


20.8 


20.4 



6 

7 

8 

9 

10 

20 

30 

40 

50 



24 


25 


2.4 


2.3 


2.8 


3.1 


3.2 


3.6 


3.5 


4.0 


3.9 


8^0 


7.8 


12.0 


11.7 


16.0 


15-6 


20.0 


19.6 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
56° 57° 



P. P. 






43 


42 


6 


4.3 


4.2 


7 


5 





4 


9 


8 


5 


7 


5 


5 


9 


6 


4 


6 


4 


10 


7 


1 


7 


1 


20 


14 


3 


14 


1 


30 


21 




21 


2 


40 


28 


g 


28 


^ 


50 


35 


8 


35 


4 





24 


^\ 


6 


2.4 


2-3 


7 


2 


8 


2 


7 


8 


3 


2 


3 


1 


9 


3 


6 


3 


5 


10 


4 





3 




20 


8 





7 


3 


30 


12 





11 


7 


40 


16 





15 


6 


50 


20 





19 


6 





23 


22 


6 


2.3 


2.i 


7 


2 


7 


2 


6 


8 


3 





3 





9 


3 


4 


3 


4 


10 


3 


g 


3 


7 


20 


7 


6 


7 




30 


11 


5 


11 


2 


40 


15 


3 


15 





50 


19 


1 


18 


7 



P. p. 



666 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
58° 59° 



Lg. Vers 



9-67217 
67240 
67263 
67285 
67308 



67331 
67354 
67376 
67399 
67422 



67445 
67467 
67490 
67513 
67535 



67558 
67581 
67603 
67626 
67649 



67671 
67694 
67717 
67739 
67762 
67784 
67807 
67830 
67852 
67875 



67897 
67920 
67942 
67965 
67987 



68010 
68032 
68055 
68077 
68100 



68122 
68145 
68167 
68190 
68212 



68235 
68257 
68280 
68303 
68324 



68347 
68369 
68392 
68414 
68436 



68459 
68481 
68503 
68526 
68548 



9.68571 
Lg. Vers. 



n 



Log.Exs, 



•94796 
.94839 
.94882 
.94925 
•94968 



•95011 
.95054 
.95097 
.95X40 
•95183 



•95226 
.95269 
.95313 
.95356 
•95399 



.95442 
.95485 
.95528 
.95571 
.95614 



•95657 
.95700 
.95744 
.95787 
•95830 



•95873 
.95916 
.95959 
.96002 
•96046 



•96089 
•96132 
.96175 
.96218 
•96261 



•96305 
.96348 
.96391 
•96434 
.9547R 



•96521 
•96564 
•96607 
•96650 
•96694 



•96737 
.96780 
.96824 
.96867 
•96910 



Log. Exs 



Lg. Vers 



9.68571 
•68593 
•68615 
•68637 
•68660 




9.69016 
•69038 
.69060 
.69082 
•69104 



9^69126 
.69149 
.69171 
.69193 
•69215 



9.69237 
•69259 
•69281 
•69303 
.69325 



9.69347 
•69369 
•69392 
•69414 
.69436 




9.69678 

•69700 

•69721 

•69743 

•69765 

9.69787 

69809 

69831 

69853 

69875 



9 69897 
Lg. Vers, 



/> 



Log. Exs. 



9.97387 
.97430 
•97473 
•97517 
•97560 




9-98038 
.98081 
•98125 
.98168 
•98211 



9.98255 
.98298 
.98342 
.98385 
•98429 



9.08472 

•98516 

•98559 

.98603 

98647 



9. 8690 
.98734 
.18777 
.98821 
.98864 



9-9^^08 
.98952 
.98:95 
.99039 
.99082 



9.99126 
•99170 
•99213 
.99257 
•99300 



9.99344 
•99388 
-99431 
•99475 
.99519 




10.00000 



Log. Exs. 





1 

2 

3 

_4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
Ik 
15 
16 
17 
18 
19 

30 

21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
33 
34_ 

35 
6 
37 
38 
39. 

40 

41 
42 
43 
44 
45 
46 
47 
48 
ii 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59. 
60 



P. P. 





44 


42 


6 


4.4 


4.3 


7 


5.1 


5.1 


8 


5^8 


5.8 


9 


6^6 


6-5 


10 


7.3 


7-2 


20 


14.6 


14.5 


30 


22.0 


21.7 


40 


29.3 


29.0 


50 


36.6 


36.2 





43 


6 


4.3 


7 


5.0 


8 


5^7 


9 


6^^: 


10 


7. , 


20 


14.3 


30 


21.5 


40 


28.6 


50 


35.^ 





33 


6 


2.3 


7 


2.7 


8 


3.0 


9 


3.4 


10 


3^8 


20 


7-6 


30 


11-5 


40 


15^3 


50 


19.1 



23 

2^2 
2^6 
3^0 
34 
3^7 
7.5 
11.2 

15. g 

18.7 



22 

2.2 
2^5 
2^9 
3.3 
3-6 
7-3 

14.6114.3 
18.3il7.9 



P. P. 



667 



TABLEVIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
60*" 61° 



Lg.Vers, 





1 
2 
3 

5 

6 

7 

8 

_9 

10 

II 

12 

13 

ii 

15 

16 

17 

18 

19 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29_ 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



40 

41 
42 
43 

44 

45 
46 
47 
48 
ii 
60 
51 
52 
53 
54 



.69897 
.69919 
.69940 
.69962 
•69984 
. 70006 
.70028 
•70050 
.70072 
.70093 



.70115 
.70137 
.70159 
.70181 
.70202 



•70224 
.70246 
•70268 
•70289 
.70311 



•70333 
.70355 
.70376 
.70398 
•7 0420 
• 70441 
•70463 
.70485 
.70507 
•70528 



•70550 
• 70 2 
.70^93 
•70615 
•7o636 



•70658 
•70680 
.70701 
.70723 
•70745 



9.70766 
.70788 
.70809 
.70831 
.70852 



9.70874 
•70896 
.70917 
.70939 
.70960 



9.70982 
.71003 
.71025 
•71046 
•71068 



9^71089 
•71111 
.71132 
.71154 
.71175 



60 9.71197 
' iLg.Vers, 



10 



10. 



10. 



Log. Exs. 



00000 
00044 
00087 
00131 
00175 



10 



00219 
00262 
00306 
00350 
00394 



00438 
00482 
00525 
00569 
00613 

00657 
00701 
00745 
00789 
00833 




10. 



01536 
01580 
01624 
01668 
01712 



10 



.01756 
.01800 
.01844 
.01889 
.01933 



10. 



01977 
02021 
02065 
02109 
02153 



10. 



02197 
02242 
02286 
02330 
02374 



Z> 



Lg. Vers. 



10. 



02418 
02463 
02507 
02551 
02595 



10-02639 



l> 1 Log. Exs. 



9.71197 
•7121C 
.712S9 
.71261 
•71282 



2> 



9 •71304 
•71325 
•71346 
•71368 
•71389 



9.71411 
.71432 
•71453 
.71475 
.71496 



1.71517 
•7153? 
•71560 
.71581 
■71603 



9.71624 

•71645 

.71667 

.71688 

^71709 

9.7173C 

^^. 71752 

II < -71773 



•71794 
71815 



9.71837 
•71858 
.71879 
.71900 
•71922 



9^71943 
.71964 
.71985 
.72006 
■72028 



9.72049 
.72070 
.72091 
.72112 
.72133 



9.72154 
.72176 
.72197 
.72218 
.72239 



9.72260 
.72281 
.72302 
.72323 
•72344 



9 •72365 
.72386 
.72408 
•72429 
•72450 



9^72471 
Lg. Vers, 



Log. Exs. 



10. 



10. 



n 



02639 
02684 
02728 
027/2 
02816 
02861 
02905 
02949 
02994 
03038 



10. 



10 



03082 
03127 
03171 
03215 
0326 
03304 
03348 
03393 
03437 
03481 



10. 



03526 
03570 
03615 
03659 
03 704 



10. 



03748 
03793 
03837 
03881 
03926 



10 



03970 
.04015 
.04059 
.041C4 
■04149 



10 



04193 
04238 
04282 
04327 
04371 



10^ 



04416 
04461 
04505 
04550 
04594 



10. 



04639 
04684 
04728 
04773 
04818 



10. 



04862 
04907 
04952 
04996 
05041 



10 



05086 
05131 
05175 
05220 
05265 



10-05310 



44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
45 
44 
44 
44 
44 
44 

44 
45 
44 
44 
44 

45 
44 
44 
44 
45 
44 
45 
44 
44 
45 

44 
45 
44 
45 
44 
45 



-O [Log. Exs. 



O 

1 

2 

3 

_4 

5 

6 

7 

8 
_9 
10 
11 
12 
13 
li 
15 
16 
17 
18 
11 
20 
21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 

35 
36 
37 
38 
li 
40 
41 
42 
43 
44 

45 
46 
47 
48 
ii 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



P.P. 





45 


43 


6 


4-5 


4^4 


7 


5.2 


5 


2 


8 


6.0 


5 


9 


9 


6^7 


6 


7 


10 


7^5 


7 


4 


20 


15^0 


14 


8 


30 


22^5 


22 


2 


40 


30.0 


29 


6 


50 


37-5 


37 


1 



44 



43 

3 
1 
8 
5 
2 
5 
7 
Q 
2 



22 21 



4 


4 


4^ 


5 


1 


5. 


5 


8 


5. 


6 


6 


6- 


7 


3 


7- 


14 


6 


14- 


22 





21- 


29 


3 


29. 


36 


6 


36. 



2 


2 


'J- 


2 


5 


2- 


2 


9 


2- 


3 


3 


3^ 


3 


6 


3 


7 


3 


7- 


11 





10^ 


14 


6 


14. 


18 


3 


17. 





21 


6 


2^1 


7 


2^4 


8 


2^8 


9 


3-1 


10 


3.5 


20 


7.0 


30 


10.5 


40 


14.0 


50 


17-5 



P.P. 



G68 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
62° 63° 



Lg. Vers. 



9.72471 
72492 
72513 
72534 
72555 



n 



72576 
72597 
72618 
72639 
72660 



72681 
72701 
72722 
72743 
72764 



72785 
72806 
72827 
72848 
72869 



72890 
72911 
72931 
72952 
72973 



72994 
73015 
73036 
73057 
73077 



73098 
73119 
73140 
73161 
73181 



73202 
73223 
73244 
73265 
73285 



73306 
73327 
73348 
73368 
73389 



73410 
7343Q 
73451 
73472 
73493 



73513 
73534 
73555 
73575 
73596 



73617 
73637 
73658 
7367? 
73699 



9-73720 
' Lg. Vers, 



Log. Exs. 



10.05310 
.05354 
.05399 
.05444 
■05489 



10.05534 
.05579 
.05623 
.05668 
.05713 



1> Lg.Vers. I> 



10.05758 
.05803 
•05848 
.05893 
.05938 



10.05983 
.06028 
.06072 
.06117 
.06162 



10.06207 
.06252 
.06297 
.06342 
.06387 



10.06432 
.06477 
.06522 
.06568 
.06613 



10.06658 
.06703 
.06748 
.06793 
.06838 



10.06883 
.06928 
.06974 
.07019 
.07064 



10.07109 
.07154 
.07200 
.07245 
.07290 



10. 07335 
.073^0 
.07426 
.07471 
.07516 



10.07562 
.07607 
.07652 
.07697 
•07743 



10.07788 
.07834 
.07879 
.07924 
.07970 



10.08015 



I> Log. Exs. 



9-73720 
73740 
73761 
73782 
73802 




74028 
74049 
74069 
74090 
74110 



74131 
74151 
74172 
74192 
74213 



74233 
74254 
74274 
74294 
74315 



74335 
74356 
74376 
74396 
74417 



74437 
74458 
74478 
74498 
74519 




74742 
74762 
74783 
74803 
74823 



74844 
74864 
7488^ 
7490^ 
74924 



74945 



Lg. Vers, 



Log. Exs. I> 



10-08015 
08061 
08106 
08151 
08197 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



08242 
08288 
08333 
08379 
08424 



08470 
08515 
08561 
08606 
08652 



08697 
08743 
0878? 
08834 
08880 



08926 
08971 
09017 
09062 
09108 



09154 
09200 
09245 
09291 
09337 



09382 
09428 
09474 
09520 
09566 



09611 
09657 
09703 
09749 
09795 



09841 
09886 
09932 
09978 
10024 



10070 
10116 
10162 
10208 
10254 



10300 
10346 
10392 
10438 
10484 



10530 
10576 
10622 
10668 
10714 



10.10760 
Log. Exs, 



P.P. 



6 


4.6 


7 


5.4 


8 


6.2 


9 


7.0 


10 


7.7 


20 


15.5 


30 


23^2 


40 


31.0 


50 


38.7 





45 


6 


4.5 


7 


5.3 


8 


6-0 


9 


6^8 


10 


7.6 


20 


15.1 


30 


22.7 


40 


30.3 


50 


37.9 



46 

4.6 

53 

6-1 

6.9 

7-6 

15.3 

23.0 

30.6 

38-3 



45 

4^5 

5.2 

6-0 

6.7 

7.5 

15.0 

22.5 

30.0 

37-5 



44 

4.4 

5-2 

5-9 

6.7 

7-4 

14.8 

22.2 

29.6 



50 37-1 



31 



2 


1 


2- 


2 


4 


2 


2 


8 


2- 


3 


1 


3 


3 


5 


3 


7 





6- 


10 


5 


10. 


14 





13. 


17 


5 


17 



20 


4 
7 

1 
4 



6 

7 

8 

9 

10 

20 

30 

40 

50 



20 

2.0 



P. P. 



669 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
64° 65° 



55 9 

56 

57 



Lg.Vers. I> 



74945 
74965 
74985 
75005 
75026 



75046 
75066 
75086 
75106 
75126 



75147 
75167 
75187 
75207 
75227 



75247 
75267 
75287 
75308 
75328 



75348 
75368 
75388 
75408 
75428 



75448 
75468 
75488 
7550§ 
75528 



75548 
75568 
75588 
75608 
75628 



75648 
75668 
75688 
75708 
75728 



75748 
75768 
75788 
75808 
75828 



75848 
75868 
75888 
75908 
75928 



75947 
75967 
75987 
76007 
76027 



76047 
76067 
76087 
76106 
76126 



76146 



' Lg. Vers. 



Log.Exs, 



10 



10760 
10807 
10853 
10899 
10945 



10- 



10991 
11037 
11084 
11130 
11176 



10 



11222 
11269 
11315 
11361 
11407 



10. 



11454 
11500 
11546 
11593 
11639 



10 



10- 



11685 
11732 
11778 
11825 
IJJ^I 

11917 
11964 
12010 
12057 
12103 



10 



10 



12150 
12196 
12243 
12289 
12336 
12383 
12429 
12476 
12522 
12569 



10. 



12616 
12662 
12709 
12756 
12802 



10 



12849 
12896 
12942 
12989 
13036 



Lg. Vers, 



10 



13083 
13130 
.13176 
13223 
.13270 



10 



13317 
13364 
.13411 
.13457 
13504 



10-13551 
Log.Exs. 



76146 
76166 
76186 
76206 
76225 



76245 
76265 
76285 
76304 
76324 
76344 
76364 
76384 
76403 
76423 



76443 
76463 
76482 
76502 
76522 



76541 
76561 
76581 
76600 
76620 



76640 
76659 
76679 
76699 
76718 

76738 
76758 
76777 
76797 
76817 



76836 
76856 
76875 
76895 
76915 



76934 
76954 
76973 
76993 
77012 



77227 
77247 
77266 
77286 
77305 



9.77325 
J> Lg.Vers. 



10 



10 



Log.Exs, 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



13551 
13598 
13645 
13692 
13739 



13786 
13833 
13880 
13927 
13974 



14021 
14068 
14115 
14162 
14210 



14257 
14304 
14351 
14398 
14445 



14493 
14540 
14587 
14634 
14682 



14729 
14776 
14823 
14871 
14918 



14965 
15013 
15060 
15108 
15155 



15202 
15250 
15297 
15345 
15392 



D 



15440 
15487 
15535 
15582 
15630 



15678 
15725 
15773 
15820 
15868 



15916 
15963 
16011 
16059 
16106 



16154 
16202 
16250 
16298 
16345 



16393 



Log. Exs. 



47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 

^1 
47 

47 

47 

47 

47 

47 

47 

47 

48 
42 

47 
47 
48 

47 
47 
48 
47 
47 
48 
47 
48 
48 
47 
48 



O 

1 
2 
3 
_4 
5 
6 
7 
8 
9 

i"o 

11 
12 
13 
14 
15 
16 
17 
18 
ii 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
-39 
40 
41 
42 
43 
j44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

60 



P.P. 



48 



4 


8 


4. 


5 


6 


5. 


6 


4 


6. 


7 


2 


7. 


8 





7. 


16 





15. 


24 





23 


32 





31 


40 





39 



47_ 

7 
5 
3 
1 
9 
8 
7 
6 
6 



47 

4-7 
5 
2 
Q 
8 



46 



4 
5 
6 
7 
7 

6 15 
5 23 
3 31 
I 38 



5 
6 
6 
7 
15 
30'23 
4030 
50 38 



46 

4.6 



61 

7, 

8 

9 

10 

20 

30 

40 

50 



20 

20 
4 
7 

1 
4 
8 
2 



2 
2 
3 
3 
6 
_ 10 
6|13 
lll6 



20 

20 



6 


1 


7 


2 


8 


2. 


9 


2. 


10 


3. 


20 


6. 


30 


9 


40 


13- 


50 


16. 



19 

9 



P. P. 



670 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
66" 67° 



Lg.Vers, 



9-77325 
77344 
77363 
77383 
77402 



77422 
77441 
77461 
77480 
77499 



J> Log.Exs. 



77519 
77538 
77557 
77577 
77596 



77616 
77635 
77654 
77674 
77693 



77712 
77732 
77751 
77770 
77790 



77809 
77828 
77847 
77867 
77886 
77905 
77925 
77944 
77963 
77982 



78002 
78021 
78040 
78059 
78078 



78098 
78117 
78136 
78155 
78174 



78194 
78213 
78232 
78251 
78270 



78289 
78309 
78328 
78347 
78366 



78385 
78404 
78423 
78442 
78462 



9-78481 



Lg.Vers, 



10. 



16393 
16441 
16489 
16537 
16585 



10 



16633 
16680 
16728 
16776 
16824 



10. 



16872 
16920 
16968 
17016 
17064 



10. 



17112 
17160 
17209 
17257 
17305 



10. 



17353 
17401 
17449 
17498 
17546 



10 



17594 
17642 
17690 
17739 
17787 



10. 



17835 
17884 
17932 
17980 
18029 



10. 



18077 
18126 
18174 
18222 
18271 



10 



18319 
18368 
18416 
18465 
18514 



10. 



18562 
18611 
18659 
18708 
18757 



10 



18805 
18854 
18903 
18951 
19000 



10. 



19049 
19098 
19146 
19195 
19244 



10-19293 



Log.Exs. 



n 



Lg. Vers 



78481 
78500 
78519 
78538 
78557 



78576 
78595 
78614 
78633 
7M52 

78671 
78690 
78709 
78728 
78747 



78766 
78785 
78804 
78823 
78842 



78861 
78880 
78899 
78918 
78937 



78956 
78975 
78994 
79013 
79032 



79051 
79069 
79088 
79107 
79126 



79145 
79164 
79183 
79202 
79220 




79427 
79446 
79465 
79484 
79503 



79521 
79540 
79559 
79578 
79596 



79615 



J> Lg.Vers, 



D 



Log. Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



19293 
19342 
19391 
19439 
1 9488 

19537 
19586 
19635 
19684 
19733 



19782 
19831 
19880 
19929 
19979 



20028 
20077 
20126 
20175 
20224 



20273 
20323 
20372 
20421 
20470 



20520 
20569 
20618 
20668 
20717 



20767 
20816 
20865 
20915 
20964 

21014 
21063 
21113 
21162 
21212 



21262 
21311 
21361 
21410 
21460 



21510 
21560 
21609 
21659 
21709 



21759 
21808 
21858 
21908 
21958 



22008 
22058 
22108 
22158 
22208 



10-22258 
Log.Exs. 



D 



D 



O 

1 

2 

3 

4 

5 

6 

7 

8 

_9 

10 

11 

12 

13 

-li 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

2g 

27 

28 

2i 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
51 

60 



P. P. 





50 


6 


5-0 


7 


5-8 


8 


6-6 


9 


7-5 


10 


8-3 


20 


16-6 


30 


25-0 


40 


33-3 


50 


41.6 



48 


4.81 


5 


6 


6 


4 


7 


2 


8 





16 





24 





32 





40 








19 


6 


1-9 


7 


2-3 


8 


2.6 


9 


2.9 


10 


3.2 


20 


6.5 


30 


9.7 


40 


13.0 


50 


16.2 



6 

7 
8 
9 
10 
20 
30 
40 
50 



1-S 

u 

2.8 

n 

9.2 
12.3 
15.4 



P.P. 



671 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT^ 
68° 69° 



Lg. Vers 



79615 
79634 
79653 
79671 
79690 



79709 
79727 
79746 
79765 
79783 



79802 
79821 
79839 
79858 
79877 



79895 
79914 
79933 
79951 
79970 



79988 
80007 
80026 
80044 
80063 



80081 
80100 
80119 
80137 
80156 



80174 
80193 
80211 
80230 
80248 



80267 
80286 
80304 
80323 
80341 



80360 
80378 
80397 
80415 
80434 



80452 
80470 
80489 
80507 
80526 



80544 
80563 
80587 
80600 
80618 



80636 
80655 
80673 
80692 
80710 
807"28 



Log.Exs. 



10.22258 
.22308 
.22358 
.22408 
.22458 



10.22508 
.22558 
.22608 
•22658 
.22708 



10-22759 
.22809 
.22859 
.22909 
.22960 



10-23010 
.23060 
.23110 
.23161 
-23211 



10.23262 
.23312 
.23362 
.23413 
.23463 



10.23514 
.23564 
.23615 
.23666 
.23716 



10-23767 
.23817 
.23868 
.23919 
-23969 



10-24020 
.24071 
.24122 
.24172 
-24223 



10-24274 
.24325 
.24376 
.24427 
.24478 



10.24529 
.24580 
.2463] 
.24682 
.24733 



10-24784 
.24835 
.24886 
.24937 
-24988 



10-25039 
.25090 
.25142 
.25193 

-25244 
10-25295 



I) 



Lg. Vers, 



9.80728 
80747 
80765 
80783 
80802 



80820 
80839 
80857 
80875 
80894 
80912 
80930 
80949 
8096Z 
80985 

81003 
81022 
81040 
81058 
81077 



81095 
81113 
81131 
81150 
81168 



81186 
81204 
81223 
81241 
81259 



81277 
81295 
81314 
81332 
81350 



81368 
81386 
81405 
81423 
81441 



81459 
81477 
81495 
81513 
81532 



81550 
81568 
81586 
81604 
81622 



81640 
81658 
81676 
81695 
81713 



81731 
81749 
81767 
81785 
8180| 
81821 



Log.Exs. 



10-25295 
25347 
25398 
25449 
25501 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



25552 
25604 
25655 
25707 
25758 



25810 
25861 
25913 
25964 
2S016 
26067 
26119 
26171 
26222 
26274 



26326 
26378 
26429 
26481 
26533 



26585 
26637 
26689 
26741 
26793 



26845 
26897 
26949 
27001 
27053 



27105 
27157 
27209 
27261 
27314 



27366 
27418 
27470 
27523 
27575 



22627 
27680 
27732 
27785 
27837 



27890 
27942 
27995 
28047 
28100 



28152 
28205 
28258 
28310 
28363 
28416 



10 

11 
12 
13 

Ji 
15 
16 
17 
18 
19 

20 
21 
22 
23 

25 
26 
27 
28 

J3 

30 
31 
32 
33 

Ji 
35 
36 
37 
38 
39 

40 
41 
42 
43 

_44 

45 
46 
47 
48 
_49 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59L 
60 



P.P. 



53 



5 


3 


5 


6 


2 


6 


7 





7 


7 


9 


7 


8 


g 


8 


17 


6 


17 


26 


5 


26 


35 


3 


35 


44 


1 


43 



52_ 

2 
I 

9 
7 
5 
2 

7 





52 


51_ 


6 


5-2 


5-1 


7 


6 





6 





8 


6 


9 


6 


8 


9 


7 


8 


7 


7 


10 


8 


6 


8 


6 


20 


17 


3 


17 


1 


30 


26 





25 


7 


40 


34 


(3 


34 


3 


50 


43 


3 


42 


9 



51 



5 


1 


5 


5 


9 


5- 


6 


8 


6 


7 


6 


7 


8 


5 


8 


17 





16 


25 


5 


25 


34 





33 


42 


5 


42. 



50 


9 
7 
6 
4 
8 
2 
6 
1 



16 
25 
40133 
50!41 



50 

5-0 
5 
6 
7 



6 
7 
8 

9 
10 
20 
30 
40 
50 



19 18 



1 


9 


1 


2 


2 


2 


2 


5 


2 


2 


3 


2 


3 


1 


3 


6 


3 


6 


9 


5 


9- 


12 


6 


12. 


15 


8 


15. 



18 
1 



"Lin^ 



ers. 



T5P 



,XS. 



Lg.Vers, 



Log.Exs. I l> 



P.P. 



672 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



Lg.Vers. 



81821 
81839 
81857 
81875 
81893 
81911 
81929 
81947 
81965 
81983 



82001 
82019 
82037 
82055 
82073 



82091 
82109 
82127 
82145 
82163 
82181 
82199 
82217 
82235 
82252 



82270 
82288 
82306 
82324 
82342 



82360 
82378 
82396 
82413 
8243] 



82449 
82467 
82485 
82503 
82520 



82538 
82556 
82574 
82592 
82609 



82627 
82645 
82663 
82681 
82698 



82716 
82734 
82752 
82769 
82787 



82805 
82823 
82840 
82858 
82876 



9.82894 
Lg. Vers 



2> 



Log. Exs, 



10.28416 
•28469 
•28521 
.28574 
•28627 




10.29476 
•29529 
•29583 
•29636 
•29689 



10.29743 
•29796 
•29850 
•29903 
•29957 



10.30010 
•30064 
•30117 
•30171 
•30225 



10 •30278 
•30332 
•30386 
•30440 
.30493 



10 • 30547 
.30601 
.30655 
.30709 
•30763 




10-31358 
•31412 
•31466 
•31521 
■31575 



2> 



10-31629 
Log. Exs. 



9.83247 
•83264 
•83282 
.83299 
•83317 



Lg.Vers 



9-82894 
•82911 
• 829'^9 
•82947 
^29^ 

9-82982 

•83000 

•83017 

83035 

^8305| 

9 •83070 
•83088 
.83106 
.83123 
^3141 

9-83159 
•83176 
•83194 
•83211 
.83229 



9 •83335 
•83352 
•83370 
•83387 
•83405 



9.83422 
•83440 
•83458 
•83475 
.83493 



9-83510 
•83528 
•83545 
•83563 
.83580 



9.83598 
•83615 
•83633 
•83650 
•83668 



9-83685 
-83703 
•83720 
-83737 
-83755 



-83772 
-83790 
-83807 
-83825 
-83842 



9-83859 
•83877 
•83894 
.83912 
■83929 



9-83946 
Lg. Vers, 



1> 



673 



Log. Exs. 



10-31629 
•31684 
•31738 
•31793 
.31847 



10.31902 
•31956 
•32011 
•32066 
-32120 



10-32175 
•32230 
•32284 
•32339 
•32394 



10 - 32449 
-32504 
•32558 
-32613 
-32668 



10.32723 
•32778 
•32833 
•32888 
-32944 



10-32999 
•33054 
•33109 
•33164 
-33220 



10.33275 
•33330 
.33385 
•33441 
•33496 



10^33552 
•33607 
•33663 
•33718 
•33774 



10-33829 
-33885 
-33941 
-33996 
.34052 





2> 



P.P. 





56 


6 


5.6 


7 


6.6 


8 


7.5 


9 


8.5 


10 


9.4 


20 


18.8 


30 


28.2 


40 


37.6 


50 


47.1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



55 



5-5 


5. 


6^5 


6- 


7-4 


7 


8.3 


8- 


9.2 


9- 


18.5 


18- 


27-7 


27- 


37.0 


36- 


46.2 


45. 



64 



5.4 


5. 


6.3 


6 


7^2 


7. 


8^2 


8. 


9.1 


9. 


18.1 


18. 


27.2 


27. 


36.3 


36- 


45.4 


45- 



56 

5-6 

6.5 

7.4 

8.4 

9.3 

18.6 

28.0 

37^3 

46.6 

55 

5 



54 

4 
3 
2 
1 










53 


53 


6 


5^3 


5-3 


7 


6^2 


6 


2 


8 


?•! 


7 





9 


8^0 


7 


9 


10 


8.9 


8 


^ 


20 


17.8 


17 


6 


30 


26.7 


26 


5 


40 


35-6 


35 


3 


50 


44.6 


44 


1 



53 

5-2 

6-1 

7.0 

7-9 

8.7 

17.5 

26.2 

35-0 

43.7 





18 


1'^ r 


6 


1 8 


1-7| 


7 


2 


1 


2 





8 


2 


4 


2 


3 


9 


2 


7 


2 


6 


10 


3 





2 


9 


20 


6 





5 


8 


30 


9 





8 




40 


12 





11 


^ 


50 


15 





14 


6 



17 

1-7 

2 
5 
8 



P.P. 



TABLE VIIT.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
72° 73° 



Lg. Vers. 1> Log.Exs. I> 



.83946 
.83964 
.83981 
.83999 
■84016 



.84033 
.84051 
.84068 
.84085 
•84103 



•84120 
•84137 
.84155 
.84172 
^84189 
.84207 
•84224 
.84241 
•84259 
•84276 



.84293 
.84310 
.84328 
.84345 

•84380 
•84397 
•84414 
•84431 
• 84449 



•84724 
•84741 
•84758 
•84775 
•84792 



•84809 
•84826 
•84844 
•84861 
•84878 



•84895 
•84912 
•84929 
•84946 
•84963 



go 9-84980 
|Lg. Vers.l 



h 10- 



10 • 



10. 



10 • 



10 • 



10 • 



10. 



36938 
36996 
37054 
37111 
37169 



10- 



10. 



37515 
37573 
37631 
37689 
37747 



10. 



37805 
37863 
37921 
37979 
38037 



10 



38095 
38153 
38212 
38270 
38328 



1038387 
Log.Exs. 



Lg. Vers, 



84980 
84997 
85014 
85031 
85049 



85066 
85083 
85100 
85117 
85134 



J> 



85151 
85168 
85185 
85202 
85219 

85236 
85253 
85270 
85287 
85304 




85575 
85592 
85608 
85625 
85642 



85659 
85676 
85693 
85710 
85726 



85743 
85760 
85777 
85794 
85811 



8582 

85844 

85861 

85878 

85895 



85911 
85928 
85945 
85962 
85979 



8^995 



Lg. Vers. 



Log. Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



38387 
38445 
38504 
38562 
38621 



38679 
38738 
38796 
38855 
38914 



38973 
39031 
39090 
39149 
39208 



39267 
39326 
39385 
39444 
39503 

39562 
39621 
39681 
39740 
39799 
39859 
39918 
39977 
40037 
40096 



40156 
40216 
40275 
40335 
40395 



40454 
40514 
40574 
40634 
40694 



40754 
40814 
40874 
40934 
40994 
41054 
41114 
41174 
41235 
41295 



41355 
41416 
41476 
41537 
41597 



41658 
41719 
41779 
41840 
41901 



41962 



^ Log, Exs. 
674 



D 



O 

1 
2 
3 
j4 

5 
6 
7 
8 
9_ 
10 
11 
12 
13 
ii 
15 
16 
17 
18 
li 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39, 
40 
41 
42 
43 
44 
45 
46 
47 
48 
ii 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



61 60 

• 

•Q 
.0 

• 1 

• 1 

• I 

• 2 
•3 
.4 



6 


1 


6^ 


7 




1- 


8 




8. 


9 




9. 


10 




10^ 


20 


3 


20 • 


30 


5 


30 • 


40 


6 


40. 


50 


8 


50. 



60 



6.0 


5 


7.0 


6. 


8.0 


7. 


9.0 


8. 


10^0 


9. 


20.0 


19 


30.0 


29 


40-0 


39^ 


50.0 


49. 



59 

9 
9 
9 
9 
9 
8 
7 
6 
6 





59 


5S 


6 


5.9 


5.8 


7 


6 


9 


6 


8 


8 


7 


8 


7 


8 


9 


8 


g 


8 


8 


10 


9 


8 


9 


7 


20 


19 


6 


19 


5 


30 


29 


5 


29 


2 


40 


39 


3 


39 





50 


49 


1 


48 


7 





58 


57 


6 


5.8 


5.7 


7 


6 


7 


6 


7 


8 


7 


7 


7 


6 


9 


8 


7 


8 


6 


10 


9 


6 


9 


6 


20 


19 


3 


19 


1 


30 


29 





28 


7 


40 


38 


6 


38 


3 


50 


48 


3 


47 


9 



57 

5.7 



20 19 
3028 
4038 
50147 



56 

5.6 





17 


17 


16 


6 


1.7 


1^7 


1^6 


7 


2.0 


2 





1 


9 


8 


2^3 


2 


2 


2 


2 


C 


2^6 


2 


5 


2 


5 


10 


2^9 


2 


8 


2 


7 


20 


5^8 


5 


g 


5 


5 


30 


8^7 


8 


5 


8 


2 


40 


11-6 


11 


3 


11 





50 


14^6 


14 


] 


i3 


7 



P. p. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

74° 75° 




9 
10 

11 
12 
13 
Ai 
15 
16 
17 
18 
19. 
20 
21 
22 
23 
24 

25 
26 
27 
28 
29. 
30 
31 
32 
33 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 
47 
48 
j49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



P.P. 





67 


6< 


5 


6 


6-7 


6.6 


7 


7.8 


7 


7 


8 


8.9 


8 


8 


9 


10.0 


10 





10 


11.1 


11 


1 


20 


22.3 


22 


1 


30 


33. 5 


33 


2 


40 


44.6 


44 


3 


50 


55.8 


55 


4 



66 

6-6 

7.7 

8.8 

9.9 

110 

22.0 

33. 

44.0 

55-0 



65 

6.5 

7.6 

8.7 

9.8 

10.9 

21.8 

32.7 

43.6 

54.6 



65 

6.5 





64 


63 


6 


6.4 


6.3 


7 


7.4 


7.4 


8 


8.5 


8.4 


9 


9.6 


9.5 


10 


10.6 


10.6 


20 


21.3 


21.1 


30 


32.0 


31.7 


40 


42.6 


42.3 


50 


53.3 


52.9 





62 


63 


6 


6.2 


6.2 


7 


7 


3 


7.2 


8 


8 


3 


8.2 


9 


9 


4 


9.3 


10 


10 


4 


10.3 


20 


20 


8 


20.6 


30 


31 


2 


31. 


40 


41 


5 


41.3 


50 


52 


1 


51.6 



61 

6-1 



10 
20 
30 
40 
50 



7 

8 

9 
10 
20 

30.5 
40.6 
50.8 



64 

6.4 

7.5 

8.6 

9.7 

10.7 

21.5 

5132.2 

3143.0 

1153.7 

63 

6.3 

7.3 

8.4 

9.4 

10.5 

21.0 

31.5 

42 

52.5 

61. 

6.1 

7.2 

8.2 

9.2 

10.2 

20.5 

30.7 

41.0 

51.2 



60 

6.0 

7.0 

8.0 

9.1 

10.1 

20.1 

30.2 

40.3 

50.4 



17 

1.7 
2.0 
2.2 
2.5 
2.8 
5 

8.5 
11.3 
14-1 



16_ 

1-6 
1 
2 

2.5 

2.7 

5.5 

8.2 

11.0 

13.7 

P. P. 



16 

1.6 
1.8 
2.1 
2.4 
2.6 
5.3 
8.0 
10.6 
13.3 



G7o 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
76° 77° 



Lg. Vers. -O Log.Exs 



87971 
87987 
88003 
88020 
88036 



88052 
88068 
88084 
88100 
88116 



88133 
88149 
88165 
88181 
88197 



88213 
88229 
88245 
88261 
88277 



88294 
88310 
88326 
88342 
88358 



88374 
88390 
88406 
88422 
88438 





88774 
88790 
88805 
88821 
88837 



88853 
88869 
88885 
8890] 
88917 



9-88933 



10.49604 
•49670 
.49737 
.49804 
.49871 



10-49939 
.50006 
.50073 
.50140 
.50208 



10.50275 
.50342 
.50410 
.50477 
• 50545 

10-50613 
.50681 
.50748 
.50816 
-50884 



10-50952 
•51020 
•51088 
•51157 
^51225 

10-51293 
•51361 
.51430 
.51498 
-51567 

10-51636 
.51704 
.51773 
.51842 
■519 11 

10.51980 
.52049 
.52118 
.52187 
-52256 



10.52325 
.52394 
•52464 
•52533 
•52603 




10-53372 
.53442 
.53512 
.53583 
-53653 



10-53724 



Lg. Vers. 



88933 
88949 
88964 
88980 
88996 



89012 
89028 
89044 
89060 
89075 



89091 
89107 
89123 
89139 
89155 



89170 
89186 
89202 
89218 
89234 




89486 
89501 
89517 
89533 
89548 



89564 
89580 
89596 
89611 
89627 
89643 
89658 
89674 
89690 
89705 



89721 
89737 
89752 
89768 
89783 



89799 
89815 
89830 
89846 
89862 



89877 



l> Log.Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



53724 
53794 
53865 
53936 
54007 



54078 
54149 
54220 
54291 
54362 



54433 
54505 
54576 
54647 
54719 



54791 
54862 
54934 
55006 
55078 



55150 
55222 
55294 
55366 
55438 



55511 
55583 
55655 
55728 
55801 

55873 
55946 
56019 
56092 
56165 



56238 
56311 
56384 
56457 
56531 



56604 
56678 
56751 
56825 
56899 



56973 
57047 
57120 
57195 
57269 



57343 
57417 
57491 
57566 
57640 
57715 
57790 
57864 
57939 
58014 



10-58089 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
9 

10 
11 
12 
13 

li 
15 
16 
17 
18 
19 

20 
21 
22 
23 

M. 
25 
26 
27 
28 

-29 

30 
31 
32 
33 
34 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 
47 
48 

j49 

50 
51 
52 
53 

_34 
55 
56 
57 
58 
59 

60 



P.P. 



75 



7-5 


7 


4 


1- 


8-7 


8 


g 


8. 


10.0 


9 


3 


9. 


11-2 


11 


\ 


10. 


12-5 


12 


3 


12. 


25-0 


24 


6 


24- 


37-5 


37 





36. 


50-0 


49 


3 


48. 


62.5 


61 


6 


60. 



73 71 



69 

6 6 
7| 8 

8 9 

9 10 
10 11 
20 23 
30 34 
40 46 
50 57 



74 73 

3 
5 
7 
9 
1 
3 
5 



7-2 


7 


1 


7 


8.4 


8 


3 


8 


9-6 


9 


4 


9 


10.8 


10 


6 


10 


12.0 


11 


8 


11 


24-0 


23 


6 


23 


36.0 


35 


5 


35 


48. 


47 


3 


47 


60.0 


59 


1 


58 



68 



g 


6 


8 


6. 





7 


g 


7. 


2 


9 





8 


3 


10 


2 


10 


.5 


11 


3 


U 


.0 


22 


6 


22 


5 


34 





33 


.0 


45 


3 


44 


.5 


56 


6 


55 



70 


2 
4 
6 
7 
3 
2 

7 



67 

7 
8 
9 

I 
3 
5 
6 

s 





66 





6 


6-6 


O^S 


7 


7 


7 








8 


8 


8 








9 


9 


9 





1 


10 


11 








]^ 


20 


22 








1 


30 


33 








2 


40 


44 








3 


50 


55 








4 





16 


16 


1 


6 


1.6 


1-6 


1. 


7 


1.9 


1 


8 


1- 


8 


2-2 


2 


1 


2. 


9 


2-5 


2 


4 


2. 


10 


2.7 


2 


6 


2. 


20 


5.5 


5 


3 


5 


30 


8.2 


8 





7. 


40 


11.0 


10 


6 


10. 


50 


13.7 


13 


3 


12- 



Lg. Vers. 



JD 



Log.Exs, 1 1> Lg. Vers 



jy 



Log.Exs. 



P. P. 



(J76 



nj 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
78° 79° 



Lg. Vers. J^ Log.Exs 



89877 
89893 
89908 
89924 
89939 



89955 
89971 
89986 
90002 
90017 
90033 
90048 
90064 
90080 
90095 



90111 
90126 
90142 
90157 
90173 



90188 
90204 
90219 
90235 
90^250 

90266 
90281 
90297 
90312 
90328 



90343 
90359 
90374 
90389 
90405 
90420 
90436 
90451 
90467 
90482 



90497 
90513 
90528 
90544 
90559 



90574 
90590 
90605 
90621 
90636 



90651 
90667 
90682 
90697 
90713 
90728 
90744 
90759 
90774 
90790 



90805 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



58089 
58164 
58239 
58315 
58390 



58465 
58541 
58616 
58692 
58768 



58844 
58920 
58995 
59072 
59148 



59224 
59300 
59377 
59453 
59530 



59606 
59683 
59760 
59837 
59914 

59991 
60068 
60145 
60223 
60300 
60378 
60455 
60533 
60611 
60688 



60766 
60844 
60923 
61001 
61079 



61158 
61236 
61315 
61393 
61472 

61551 
61630 
61709 
61788 
61867 



61947 
62026 
62105 
62185 
62265 



10-62345 

62424 

62504 

.62585 

.62665 

10.62745 

Log.Exs.l 




P. P. 



86 85 



8 


6 


8 


5 


8 


10 





9 


9 


9 


11 


4 


11 


3 


11 


12 


9 


12 


7 


12 


14 


3 


14 


1 


14 


28 


6 


28 


3 


28 


43 





42 


5 


42 


57 


3 


56 


6 


56. 


71 


6 


70 


8 


70 



84 
4 
8 
2 
6 








83 
8-3 



11 
12 
13 
27 
30|41 
40 55 
50J69 



7 9 
10 
412 
8|13 
6 27 
541 
3i54 
ll68 



82 
82 



81 

8 

9 
10 
12 
13 
27 
40 
54 
67 



80 79 



8 


0, 7 


9 


7 


9 


3| 9 


2 


9 


10 


6 10 


5 


10 


12 


11 


8 


11. 


13 


3 13 


1 


13- 


26 


6 26 


3 


26- 


40 


39 


5 


39- 


53 


3 52 


6 


52. 


66 


6 65 


8 


65. 





77 


76 


7.^ 


6 


7-7 


7.6 


7 


7 


9 





8 


8 


8 


8 


10 


2 


10 


1 


10 


9 


11 


5 


11 


4 


11 


10 


12 


8 


12 


6 


12 


20 


25 


6 


25 


3 


25 


30 


38 


5 


38 





37 


40 


51 


3 


50 




50 


50 


64 


1 


63 


3 


62 



78 
8 
1 
4 
7 








40 
50!0 



16 



6 


1 


6 


1 


7 


1 


8 


1 


8 


2 


1 


2 


9 


2 


4 


2 


10 


2 


6 


2 


20 


5 


3 


5 


30 


8 





7 


40 


10 


6 


10 


50 


13 


3 


12 



15_ 

5 
8 

3 
6 
I 
7 
3 

± 

P. P. 



15 

1-5 



677 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
80° 81° 



Lg. Vers, 



9.91716 
91731 
91746 
91761 
91776 



91791 
91807 
91822 
91837 
91852 



91867 
91882 
91897 
91912 
91927 



91942 
91957 
91972 
91987 
92002 



92016 
92031 
92046 
92061 
92076 



92091 
92106 
92121 
92136 
92151 



92166 
92181 
92196 
92211 
92226 



92240 
92255 
92270 
92285 
92300 



92315 
92330 
92345 
92360 
92374 



92389 
92404 
92419 
92434 
92449 



92463 
92478 
92493 
92508 
92523 



92538 
92552 
92567 
92582 
92597 
92612 



' Lg. Vers, 



Log.Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



67749 
67836 
67923 
68010 
68097 
68184 
68272 
68359 
68447 
68534 
68622 
68710 
68798 
68886 
68975 



n 



69063 
69152 
69240 
69329 
69418 



69507 
69596 
69686 
69775 
69865 



69955 
70044 
70134 
70224 
70315 



70405 
70495 
70586 
70677 
70768 
70859 
70950 
71041 
71133 
71224 



71316 
71408 
71500 
71592 
71684 



71776 
71869 
71961 
72054 
72147 



72240 
72333 
72427 
72520 
72614 



72707 
72801 
72895 
72990 
73084 



10-73178 
Log.Exs. 



Lg. Vers. 



92612 
92626 
92641 
92656 
92671 



92686 
92700 
92715 
92730 
92745 



92759 
92774 
92789 
92804 
92818 



92833 
92848 
92862 
92877 
92892 



92907 
92921 
92936 
92951 
92965 



92980 
92995 
93009 
93024 
93039 
93051 
93068 
93083 
93097 
93112 
93127 
93141 
93156 
93171 
93185 



93200 
93214 
93229 
93244 
93258 



93273 
93287 
93302 
93317 
93331 
93346 
93360 
93375 
93389 
93404 



93419 
93433 
93448 
93462 
93477 



9-93491 
Lg. Vers. 



Log. Exs. J> 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



73178 
73273 
73368 
73463 
73558 



73653 
73748 
73844 
73940 
74035 



74131 
74227 
74324 
74420 
74517 
74613 
74710 
74807 
74905 
75002 



75099 
75197 
75295 
75393 
75491 



75589 
75688 
75786 
75885 
75984 

76083 
76182 
76282 
76382 
76481 
76581 
76681 
76782 
76882 
76983 



77083 
77184 
77286 
77387 
77488 



77590 
77692 
77794 
77896 
77998 
78101 
78203 
78306 
78409 
78513 



78616 
78720 
78823 
78927 
79031 



79136 



^ Log. Exsi 
678 



95 

94 

95 

95 

95 

95 

95 

96 

95 

96 

96 

96 

96 

96 

96 

97 

97 

97 

97 

97 

98 

97 

98 

98 

98 

98 

98 

99 

99 

99 

99 

99 

100 

99 

100 

100 

100 

100 

100 

100 

101 

101 

101 

101 

101 

102 

102 

102 

102 

102 

102 

103 

103 

103 

103 

104 

103 

104 

104 

104 





1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
ii 
15 
16 
17 
18 
2i 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
51 
55 
56 
57 
58 
59. 
60 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



90 

9.0 





9 


6 


0.9i 


7 


1 





8 


1 


2 


9 


1 


3 


10 


1 


5 


20 


3 





30 


4 


5 


40 


6 





50 


7 


5 



6 

7 

8 

9 

10 

20 

30 

40 

50 






7 





8 








1 





1 


1 


2 


3 


3 


5 


4 


6 


5 


8 



6 





5 


7 





6 


8 





6 


9 





7 


10 





8 


20 


1 


6 


30 


2 


5 


40 


3 


3 


50 


4 


1 



15_ 



9 
10 
20 
30 
40 
50 



1 


5 


1 


8 


2 


6 


2 


3 


2 


6 


5 




7 


7 


10 


3 


12 


9 



6 

7 

8 

9 
10 
20 
30 
40 
50 

P. P. 



80 
8.0 
9.3 
10.6 
12-0 
13.3 
26.6 
40.0 
53.5 
66.6 

8 
0.8 
0.9 
l.Q 
1.2 
1.3 
2.6 
4.0 
5-3 



6 

0.6 
0.7 
0.8 
0.9 
1.0 
2.0 
3.0 
4.0 
5.0 

4 

0.4 
0.4 
0.5 
0.6 
0.6 
1.3 
2.0 
2.6 
3.3 

15 

1.5 
1.7 
2.0 
2.2 
2.5 
5.0 
7.5 
10.0 
12.5 

1? 

1.4 
1-7 
1.9 
2.2 
2.4 
4.8 
7.2 
9.6 
12.1 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

82° 83° 



' Lg. Vers 



9.93491 
93506 
93520 
93535 
93549 



93564 
93578 
93593 
93607 
93622 



93636 
93651 
93665 
93680 
93694 



93709 
93723 
93738 
93752 
93767 



93781 
93796 
93810 
93824 
93839 



93853 
93868 
93882 
93897 
93911 



93925 
93940 
93954 
93969 
93983 



93997 
94012 
94026 
94041 
94055 



94069 
94084 
94098 
94112 
94127 
94141 
94155 
94170 
94184 
94198 



94213 
94227 
94241 
94256 
94270 



94284 
94299 
94313 
94327 
94341 



9-94356 
' Lg.Vers 



10 



Log. Exs. D Lg.Vers 



10-79136 
79240 
79345 
79450 
79555 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



79660 
79766 
79871 
79977 
80083 



80189 
80296 
80402 
80509 
80616 
80723 
80831 
80988 
81046 
81154 



81262 
81371 
81479 
81588 
81697 



81806 
81916 
82025 
82135 

82245 



82356 
82466 
82577 
82688 
82799 



82910 
83022 
83133 
83245 
83358 



83470 
83583 
83695 
83809 
83922 
84035 
84149 
84263 
84377 
84492 
84607 
84721 
84837 
84952 
85068 



85183 
85299 
85416 
85532 
85649 
85766 



Log. Exs. 



104 

105 

104 

105 

105 

105 

105 

106 

106 

106 

106 

106 

107 

107 

107 

107 

107 

108 

108 

108 

108 

108 

109 

109 

109 

109 

109 

110 

110 

110 

110 

110 

11 

11 

11 

11 

11 

112 

112 

112 

112 

112 

113 

113 

113 
114 
114 
114 
114 

115 
114 
115 
115 
116 
115 
116 
116 
116 
117 
117 



9.94356 
94370 
94384 
94398 
94413 



94427 
94441 
94456 
94470 
94484 



94498 
94512 
94527 
94541 
94555 



94569 
94584 
94598 
94612 
94626 



94640 
94655 
94669 
94683 
94697 



94711 
94726 
94740 
94754 
94768 



94782 
94796 
94810 
94825 
94839 



94853 
94867 
94881 
94895 
94909 




95064 
95078 
95093 
95107 
95121 



95135 
95149 
95163 
95177 
95191 



9.95205 
Lg. Vers. 



D Log. Exs. n 



10-85766 
85884 
86001 
86119 
86237 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



86355 
86474 
86592 
86711 
86831 



86950 
87070 
87190 
87310 
87431 



87552 
87673 
87794 
87916 
88038 



88160 
88282 
88405 
88528 
88651 



88775 



89022 
89147 
89271 



89396 
89521 
89647 
89773 
89899 



90025 
90152 
90279 
90406 
90533 



90661 
90789 
90917 
91046 
91175 



91304 
91434 
91564 
91694 
91825 



91956 
92087 
92218 
92350 
92482 
92614 
92747 
92880 
93014 
93147 



10-93281 



117 
117 
117 
118 
118 
118 
118 
119 
119 
119 
120 
120 
120 
120 
121 
121 
121 
121 
122 
122 
122 
122 
123 
123 
124 
123 
124 
124 
124 
125 
125 
125 
126 
126 
126 
126 
127 
127 
127 
128 
127 
128 
129 
129 
129 
130 
129 
130 
130 

131 
131 
131 
131 
132 

132 
133 
133 
133 
133 
134 



Log. Exs. 



o 

1 

2 
3 

_4 

5 
6 
7 
8 
_9 

10 
11 
12 
13 
li 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 

31 

30 
31 
32 
33 

-M 
35 
36 
37 
38 

M, 

40 
41 
42 
43 

_44 
45 
46 
47 
48 

-ii 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59_ 

60 



P.P. 





130 


6 


13.0 


7 


15.1 


8 


17.3 


9 


19.5 


10 


21.6 


20 


43.3 


30 


65.0 


40 


86.6 


50 


108.3 





110 


6 


11.0 


7 


12.8 


8 


14.6 


9 


16.5 


10 


18-3 


20 


36.6 


30 


55.0 


40 


73-3 


50 


91.6 





3 


6 


O.J 


7 


O.J 


8 


0.4 


9 


0.4 


10 


0.5 


20 


1.0 


30 


1.5 


40 


2.0 


50 


2.5 





1 


6 


0.1 


7 





1 


8 





1 


9 





\ 


10 





\ 


20 





3 


30 





5 


40 







50 





8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



15 

1.4 
1.7 
1.9 
2.2 
2.4 
4.8 
7.2 
9.6 
12.1 



120 

12.0 
14.0 
16.0 
18.0 
20.0 
40.0 
60.0 
80.0 
100.0 



100 

10.0 
11.6 
13-3 
15.0 
16.6 
33.3 
50.0 
66.6 
83.3 



2 

0.2 
0.2 
0.2 
0.3 
0.3 
0.6 
10 
1.3 
1.6 



O 

o.i 

0.0 
0.0 
0.1 
0-1 
O.I 
0.2 
03 
0.4 



14 

1.4 
1.6 
1.8 
2.1 
2.3 
4.6 
7.0 
9.3 
11.6 



P.P. 



679 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 

84'=' 85° 



Lg.Vers. 



O 

1 
2 
3 

5 

6 

7 

8 

_9_ 

10 

11 

12 

13 

li 

15 

16 

17 

18 

19 



20 

21 
22 
23 
24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

33 

39 

40 

41 

42 

43 

44 



45 
46 
47 
48 



I) 



.95205 
.95219 
.95233 
.95247 
.95261 



•95275 
•95289 
.95303 
•95317 
95331 



•95345 
•95359 
•95373 
•95387 
95401 



•95415 
•95429 
•95443 
• 95457 
95471 



9. 95485 
•95499 
•95513 
•95527 
•95540 

995554 
•95568 
■95582 
•95596 
95615 



9 95624 
95638 
95652 
95666 
•95680 



9-95693 
•95707 
•95721 
•95735 
•95749 



995763 
•95777 
•95791 
•95804 
•95818 



9. 95832 

•95846 

•95860 

95874 



49 95888 



50 

51 

52 

53 

54 

55 

56 

57 

58 

59_ 

60 



9^95901 
•95915 
.95929 
.95943 
.95957 



9.95970 
•95984 
•95998 
•96012 
•96026 



9 •96039 
Lg. Vers 



Log.Exs. 



10 



93281 
93416 
93551 
93686 
93821 



10 



93957 
94093 
94229 
94366 
94503 



10 



10 



10 



10 



10 



10 



10 



10 



11 



11 



11 



94641 
94778 
94917 
95055 
95194 



95333 
95473 
95613 
95753 
95894 



96035 
96176 
96318 
96461 
96603 
96746 
96889 
97033 
97177 
97322 



97467 
97612 
97758 
97904 
98050 



98197 
98345 
98492 
9??640 
98789 



98938 
99087 
99237 
99387 
99538 



99841 
99993 
00145 
00298 



1> Lg.Vers. 



00451 
00605 
00759 
00914 
01069 



01225 
01381 
01537 
01694 
01852 
02010 



134 
135 
135 
135 
135 
136 
136 
137 
137 
137 
137 
138 
138 
139 
139 
139 
140 
140 
140 
14l 
141 
142 
142 
142 
143 
143 
144 
144 
144 
145 
145 
145 
146 
146 
147 
147 
147 
148 
149 
149 
149 
150 
150 
151 
151 
15l 
152 
152 
153 
153 
154 
154 
155 
155 
155 
156 
156 
157 
157 
158 



Log.Exs. 



•96039 
.96053 
•96067 
•96081 
:i6095 
•96108 
•96122 
•96136 
•96150 
•96163 



•96177 
•96191 
•96205 
•96218 
•96232 



•96246 
•96259 
•96273 
•96287 
•96301 



•96314 
•96328 
•96342 
•96355 
•96369 



96383 
•96397 
•96410 
•96424 
■96438 
•96451 
•96465 
.96479 
.96492 
•96506 
•96519 
•96533 
■96547 
•96560 
•96574 



96588 
•96601 
•96615 
•96629 
•96642 



•96656 
•96669 
•96683 
•96697 
•96710 



•96724 
•96737 
•96751 
•96764 
•96778 



96792 
•96805 
•96819 
•96832 
•96846 



996859 



D Lg.Vers 



2> 



Log.Exs. 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



02010 
02168 
02327 
02487 
0^2646 
02807 
02968 
03129 
03291 
03453 



03616 
03780 
03944 
04108 
04273 



04438 
04604 
04771 
04938 
05106 



05274 
05443 
05612 
05782 
05952 



06123 
06295 
06467 
06640 
06813 



06987 
07161 
07336 
07512 
07688 



07865 
08043 
08221 
08400 
08579 



08759 
08940 
09121 
09303 
09486 



09669 
09853 
10038 
10223 
10409 



10595 
10783 
10971 
11160 
11349 



11539 
11730 
11922 
12114 
12307 



J> 



12501 



Log.Exs. 



158 
159 
159 
159 
160 
161 
161 
161 
162 
163 
163 
164 
164 
165 
165 
166 
167 
167 
167 
168 
169 
169 
169 
170 
171 
171 
172 
173 
173 
174 
174 
175 
176 
176 

177 
177 
178 
179 
179 
180 
180 
181 
182 
182 

183 
184 
185 
185 
186 
186 
187 
188 
189 
189 
190 
191 
191 
192 
193 
193 
~ 





1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 
20 
21 
22 
23 
24 

25 
26 
27 
28 
li 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

55 
56 
57 
58 

60 



P. P. 



7 
8 
9 
10 
20 
30 
40 
50 



10 
20 
30 
40 
50 



190 



19 





22 


1 


25 


3 


28 


5 


31 


6 


63 


3 


95 





126 


6 


158 


3 



170 



17 





16. 


19 


8 


18^ 


22 


6 


21^ 


25 


5 


24 • 


28 


3 


26^ 


56 


6 


53 • 


85 





80 • 


113 


3 


106 


141 


6 


133 



150 



15 





14^ 


17 


5 


16^ 


20 





18^ 


22 


5 


21^ 


25 





23 


50 





46 


75 





70 


100 





93 


125 





116 



180 

18-0 
21.0 
24.0 
27.0 
30.0 
60.0 
90.0 
120.0 
150.0 

160 



6 
3 

6 
3 

6 
3 

140 







130 


9 


? 


6 


13. 


0.9 


0. 


7 


15 


1 


1-0 


0^ 


8 


17 


3 


1-2 


1^ 


9 


19 


5 


1.3 


1. 


10 


21 




1.5 


1^ 


20 


43 


3 


30 


2- 


30 


65 





4.5 


4 


40 


86 


6 


60 


5^ 


50- 


108 


3 


7^5 


6 



7 

60^7 

7i08 

80-9 

9 10 

1011 

202^3 

30 3-5 

40'4-6 

505.8 



6 

0.6 
07 
0^8 
0^9 
1^0 
2^0 
3-0 
4^0 
5^0 



5 

05 



14 

1.4 
1-7 
1.9 
2.2 
2^4 
4-8 
7-2 



12^1 



14 


li 


1^4 


1- 


16 


1. 


1-8 


1. 


2^1 


2. 


2^3 


2- 


4^6 


^^ 


7^0 


6. 


9^3 


9. 


11^6 


11^ 



P. p. 



680 



, TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

86° 87° 



Lg. Vers. 



9.96859 
96837 
96887 
96900 
96914 



9.96927 
•96941 
.96954 
.96968 
•96981 



9 •96995 
97008 
97022 
97035 
97049 



•97062 
•97076 
•97089 
•97103 
•97116 



•97130 
•97143 
•97157 
•97170 
•97183 



•97197 
•97210 
•97224 
•97237 
•97251 



9^97264 
•97277 
•97291 
•97304 
•97318 



9^97331 
•97345 
•97358 
•97371 
•97385 



9. 97398 
97412 
97425 
97438 
97452 



9.97465 
97478 
97492 
97505 
97519 



9-97532 
•97545 
•97559 
•97572 
•97585 



9^97599 
•97612 
•97625 
•97639 
.97652 



9.97665 



Lg.Vers. 



Log. Exs. 



11.12501 
•12696 
•12891 
•13087 
•13284 



11^13482 
•13680 
•13879 
•14079 
■14280 



11^14482 
.14684 
14887 
15092 
15297 



11^15502 
15709 
15917 
16125 
16334 



2> 



J> 



11^16544 
16755 
16967 
17180 
17394 



11^17609 
•17824 
•18041 
•18259 
•18477 



11^18697 
•18917 
.19138 
.19361 
•19584 



11-19809 
.20034 
•20261 
.20489 
.20717 



11.20947 
21178 
•21410 
21643 
21877 



11^22112 
•22349 
•22586 
•22825 
• 23^65 
11 •23306 
23548 
23792 
24037 
24283 



11^24530 
•24778 
.25028 
.25279 
.25531 



11.25785 



195 

195 

196 

196 

198 

198 

199 

200 

201 

201 

202 

203 

204 

205 

205 

206 

208 

208 

209 

210 

211 

212 

213 

214 

214 

215 

216 

218 

218 

219 

220 

221 

222 

223 

224 

225 

227 

227 

228 

230 

230 

232 

233 

234 

235 

236 

237 

239 

23^ 

241 

242 

243 

245 

246 

247 
248 
250 
251 
252 
254 



Lg. Vers. 



1.97665 
.97679 
•97692 
•97705 
97718 



•97732 
•97745 
•97758 
•97772 
•97785 



•97798 
-97811 
•97825 
•97838 
•97851 



• 9786 

•97878 

•97891 

•97904 

•97917 



Log. Exs. I D Lg.Vers. 



•97931 
•97944 
.97957 
.97970 
.97984 



.97997 
.98010 
.98023 
.98036 
•98050 



•98063 
•98076 
.98089 
.98102 
•98116 



.98129 
.98142 
.98155 
.98168 
.98181 




9. 



98326 
98339 
98352 
98365 
98378 



98392 
98405 
98418 
98431 
98444 



9-98457 



Log. Exs, 



11.25785 
26040 
26297 
26554 
26814 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



27074 
27336 
27599 
27864 
28131 



28398 
28668 
28938 
29211 
29485 



29760 
30037 
30316 
30596 
30878 



31162 
31447 
31734 
32023 
32313 



32606 
32900 
33196 
33494 
33793 
34095 
34398 
34704 
35011 
35321 



D 



35632 
35946 
36261 
36579 
36899 



37221 
37546 
37872 
38201 
38532 



38866 
39201 
39540 
39880 
40224 



40569 
40918 
41269 
41622 
41979 



42338 
42699 
43064 
43431 
43802 



44175 



255 
256 
257 
259 
260 
262 
263 
265 
266 
267 
269 
270 
272 
274 
275 
277 
278 
280 
282 
283 
285 
287 
288 
290 
292 
294 
296 
298 
299 
301 
303 
305 
307 
309 
311 
313 
315 
318 
320 
322 
324 
326 
328 
331 
333 
335 
338 
340 
343 
345 
348 
351 
353 
356 
359 
361 
364 
367 
370 
373 



Log. Exs. 



2> 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 
10 
11 
12 
13 
ii 
15 
16 
17 
18 
_19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39_ 
40 
41 
42 
43 

45 
46 
47 
48 
49_ 
50 
51 
52 
53 
54 
55 
56 
57 
58 

60 



P. P. 





250 


6 


25.0 


7 


29.1 


8 


33.3 


9 


37.5 


10 


41.6 


20 


83.3 


30 


125.0 


40 


166.6 


50 


208.3 





330 


6 


23.0 


7 


26.8 


8 


30.6 


9 


34.5 


10 


38.3 


20 


76.6 


30 


115-0 


40 


153^3 


50 


191.6 



10 
20 
30 
40 
50 



210 

21.0 

24.5 

28.0 

31-5 

35^0 

70^0 

105-0 

140.0 

175-0 





190 


4 


6 


19^0 


0.4 


7 


22 


\ 


0.4 


8 


25 


3 


0.5 


9 


28 


5 


0.6 


10 


31 


5 


0.6 


20 


63 


8 


1.3 


30 


95 





2.0 


40 


126 


5 


2.6 


50 


158 


3 


3.3 





2 


1 


6 


0.2 


0-1 


7 


0^2 


0^: 


8 


0^2 


0. 


9 


0^3 


0^ 


10 


0^3 


0^ 


20 


0^6 


0^8 


30 


1^0 


0^5 


40 


1^3 


0^6 


50 


1^6 


0^8 



14 

1^4 
1-6 
1^8 
2-1 
2-3 
4-6 
7^0 
9^3 
]1^6 



13_ 

1^3 
1.6 
1.8 
2.0 
2.2 
4.5 
6^7 
9^0 
11-2 



240 

24.0 

28.0 

32.0 

36.0 

40.0 

80.0 

120.0 

160.0 

200.0 

220 

22.0 

25.6 

29.3 

33.0 

36.6 

73.3 

110.0 

146.6 

183.3 

200 

20.0 

23.3 

26.6 

30.0 

33.3 

66.6 

100. g 

133.3 

166.6 



3 

0.3 
0.3 
0.4 
0.4 
05 
1.0 
1.5 
2.0 
2.5 

O 

0.5 
0.0 
0.0 
0.1 
0.1 
0-1 
0^2 
0.3 
0.4 

13 

1.3 
1.5 
1.7 
1.9 
3.1 
4.3 
6.5 
8.6 
10^8 



'. P. 



681 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANT? 
88** 89** 

P. P. 




15 



1.3 


1. 


1.6 


1. 


1.8 


1. 


2.0 


1. 


2.2 


2. 


4.5 


4. 


6.7 


6. 


9.0 


8. 


11.2 


10. 



13 
3 



6 

7 

8 

9 

10 

20 

30 

40 

50 



12 

1.2 
1.4 
1.6 
1.9 
2.1 
4.1 
6.2 
8-3 
10.4 



P.P. 



682 



TABLE IX.— NATURAL SINES, COSINES. TANGENTS, AND COTANGENTS. 









0" 








!*> 








Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


f 




1 

2 
3 
4 


.00000 
.00029 
.00058 
.00087 
.00116 


One 
One 
One 
One 
One 


.00000 
.00029 
.00058 
.00087 
.00116 


Infinite 
3437.75 
1718.87 
1145.92 
859.436 


.01745 
.01774 
.01803 
.01832 
.01862 


.99985 
.99984 
.99984 
.99983 
.99983 


.01746 
.01775 
.01804 
.01833 
.01862 


57.2900 
56.3506 
55.4415 
54.5613 
53.7086 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.00145 
.00175 
.00204 
.00233 
.00262 


One 
One 
One 
One 
One 


.00145 
.00175 
.00204 
.00233 
.00262 


687.549 
572.957 
491.106 
429.718 
381.971 


.01891 
.01920 
.01949 
.01978 
.02007 


.99982 
.99982 
.99981 
.99980 
.99980 


.01891 
.01920 
•01949 
•01978 
•02007 


52.8821 
52^0807 
51 •3032 
50.5485 
49.8157 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.00291 
.00320 
.00349 
.00378 
.00407 


One 
.99999 
.99999 
.99999 
•99999 


.00291 
.00320 
.00349 
.00378 
.00407 


343.774 
312^521 
286.478 
264.441 
245.552 


.02036 
.02065 
.02094 
.02123 
.02152 


•99979 
•99979 
•99978 
•99977 
•99977 


•02036 
•02066 
•02095 
•02124 
.02153 


49.1039 
48.4121 
47.7395 
47.0853 
46.4489 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.00436 

.00465 
.00495 
.00524 
.00553 


.99999 
.99999 
.99999 
.99999 
.99998 


.00436 
.00465 
.00495 
.00524 
.00553 


229.182 
214.858 
202.219 
190.984 
180.932 


.02181 
.02211 
.02240 
•02269 
.02298 


=99976 
•99976 
.99975 
•99974 
.99974 


.02182 
.02211 
.02240 
.02269 
•02298 


45.8294 
45.2261 
44.6386 
44.0661 
43.5081 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.00582 
.00611 
•00640 
.00669 
.00698 


.99998 
.99998 
.99998 
.99998 
.99998 


.00582 
.00611 
.00640 
.00669 
.00698 


171. 885 
163.700 
156.259 
149.465 
143.237 


.02327 
.02356 
.02385 
.02414 
.02443 


•99973 
•99972 
•99972 
•99971 
•99970 


•02328 
.02357 
-02386 
•02415 
.02444 


42.9641 
42.4335 
41.9158 
41.4106 
40.9174 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.00727 
.00756 
.00785 
.00814 
.00844 1 


.99997 
.99997 
.99997 
.99997 
.99996 


.00727 
.00756 
.00785 
.00815 
.00844 


137-507 
132-219 
127.321 
122.774 
118.540 


.02472 
.02501 
.02530 
.02560 
.02589 


.99969 
.99969 
.99968 
•99967 
.99966 


.02473 
.02502 
.02531 
.02560 
.02589 


40.4358 
39.9655 
39-5059 
39-0568 
38.6177 


35 
34 
33 
32 
31 


30 

31 
32 
33 

34 


.00873 1.99996 
.00902 1.99996 
.00931 .99996 
.00960 j. 99993 
.00989 '.99995 


.00873 
.00902 
.00931 
.00960 
.00989 


114.589 
110.892 
107.426 
104.171 
101.107 


.02618 .99966 
.02647 .99965 
•02676 .99964 
.02705 .99963 
.02734 .99963 


.02619 
.02648 
.02677 
.02706 
.02735 


38.1885 
37.7686 
37-3579 
36-9560 
36.5627 


30 

29 
28 

27 
26 


35 
36 
37 
38 
39 


.01018 .99995 
.01047 .99995 
.01076 .99994 
.01105 .99994 
•01134 .99994 


.01018 
.01047 
.01076 
.01105 
.01135 


98.2179 
95.4895 
92.9085 
90-4633 
88. 1436 


.02763 .99962 
.02792 .99961 
.02821 .99960 
.02850 |. 99959 
.02879 .99959 


.02764 
.02793 
.02822 
.02851 
.02881 


36.1776 

35.8006 

35.4313^ 

35.0695 

34.7151 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.01164 
.01193 
.01222. 
.01251 
.01280 1 


.99993 
.99993 
•99993 
.99992 
•99992 


.01164 
.01193 
.01222 
.01251 
.01280 


85- 9398 
83- 8435 
81.8470 
79.9434 
78. 1263 


•02908 .99958 \ 
•02938 .99957 
.02967 .99956 
.02996 .99955 
.03025 (.99954 


•02910 
.02939 
.02968 
.02997 
.03026 


34.3678 
34.0273 
33.6935 
33-3662 
33-0452 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.01309 i 

.01338 

.01367 

.01396 

.01425 


•99991 
•99991 
.99991 
.99990 
.99990 

.99989 
.99989 
.99989 
.99988 
.99988 


.01309 
.01338 
.01367 
.01396 
.01425 


76.3900 
74-7292 
73-1390 
71-6151 
70-1533 


03054 !. 99953 
03083 1.99952 i 
.03112 |. 99952 | 
.03141 .99951 
•03170 .99950 | 


-03055 
•03084 
•03114 
•03143 
.03172 


32.7303 
32-4213 
32-1181 
31-8205 
31-5284 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.01454 
.01483 
.01513 
•01542 
.01571 


.01455 
.01484 
.01513 
.01542 
.01571 


68-7501 
67-4019 
66-1055 
64-8580 
63.6567 


.03199 
.03228 
.03257 
•03286 
.03316 


.99949 
.99948 
.99947 
.99946 
.99945 i 


.03201 
-03230 
-03259 
-03288 
.03317 


31.2416 
30-9599 
30-6833 
30^4116 
30 • 1446 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.01600 
.01629 

01658 
.01687 

01716 


.99987 
.99987 
.99986 
.99986 
.99985 


.01600 
.01629 
•01658 
•01687 
.01716 


62.4992 
61.3829 
60-3058 
59.2659 
58.2612 


.03345 
.03374 
•03403 
.03432 
•03461 


.99944 
•99943 
.99942 
.99941 
•99940 
•99939 


•03346 
•03376 
•03405 
.03434 
.03463 


29 •8823 
29^6245 
?9-3711 
za.l220 
28.8771 


5 

4 
3 

2 

1 


60 


.01745 


.99985 


.01746 


57-2900 


•03490 


.03492 


28.6363 
Tan. 





' 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 1 Sin. j Cot. 


/ 



89** 



683 



88^ 



TABLE IX.- 



-NATURAL SINES. COSINES, TANGENTS, AND COTANGENTS, i 
3° 3^ 



t 


Sin. 


Cos. 1 Tan. 


Cot. 


Sin. 


Cos. 


Tan. 1 Cot. ' 


/ 




1 

2 
3 
4 


.03490 
.03519 
.03548 
.03577 
.03606 


.99939 
.99938 
.99937 
.99936 
.99935 


.03492 
.03521 
.03550 
.03579 
.03609 


28.6363 
28.3994 
28.1664 
27.9372 
27.7117 


.05234 
.05263 
.05292 
.05321 
.05350 


.99863 
.99861 
.99860 
.99858 
.99857 


.05241 
.05270 
.05299 
.05328 
.05357 


19 
18 
18 
18 
18 


0811 
9755 
8711 
7678 
6656 


60 

59 
58 
57 
56 


5 

6 
7 
8 
9 


.03635 
.03664 
.03693 
.03723 
.03752 


.99934 
.99933 
.99932 
.99931 
.99930 


.03638 
.03667 
.03696 
.03725 
.03754 


27.4899 
27.2715 
27.0566 
26.8450 
26.6367 


.05379 
-05408 
.05437 
.05466 
•05495 


.99855 
.99854 
.99852 
.99851 
.99849 


.05387 
.05416 
.05445 
.05474 
.05503 


18 
18 
18 
18 
18 


5645 
4645 
3655 
2677 
1708 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.03781 
.03810 
.03839 
.03868 
.03897 


.99929 
.99927 
.99926 
.99925 
.99924 


.03783 
.03812 
.03842 
.03871 
.03900 


26.4316 
26-2296 
26.0307 
25.8348 
25.6418 


-05524 
.05553 
-05582 
.05611 
.05640 


.99847 
.99846 
.99844 
.99842 
.99841 


.05533 
.05562 
.05591 
.05620 
.05649 


18 
17 
17 
17 
17 


0750 
9802 
8863 
7934 
7015 


50 

49 
48 
47 
48 


15 
16 
17 
18 
19 


.03926 
.03955 
.03984 
.04013 
.04042 


.99923 
.99922 
.99921 
.99919 
.99918 


.03929 
.03958 
.03987 
.04016 
.04046 


25.4517 
25.2644 
25.0798 
24.8978 
24.7185 


.05669 
.05698 
-05727 
.05756 
•05785 


.99839 
.99838 
.99836 
.99834 
•99833 


.05678 
.05708 
-05737 
.05766 
.05795 


17 
17 
17 
17 
17 


6106 
5205 
4314 
3432 
2558 


45 
44 
43 
42 

^ 


30 

21 
22 
23 
24 


.04071 
.04100 
.04129 
.04159 
.04188 


.99917 
.99916 
.99915 
.99913 
.99912 


.04075 
.04104 
.04133 
•04162 
•04191 


24.5418 
24-3675 
24.1957 
24.0263 
23 .8593 
23-6945 
23-5321 
23-3718 
23-2137 
23-0577 
22-9038 
22-7519 
22-6020 
22-4541 
22^3081 


-05814 
.05844 
•05873 
.05902 
.05931 
.05960 
-05989 
.06018 
.06047 
.06076 


•99831 
•99829 
•99827 
•99826 
•99824 
.99822 
.99821 
.99819 
-99817 
•99815 


-05824 
.05854 
.05883 
.05912 
•05941 


17 
17 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 


1693 
0837 
9990 
9150 
8319 
7496 
6681 
5874 
5075 
4283 
3499 
2722 
1952 
1190 
0435 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.04217 
.04246 
.04275 
.04304 
.04333 


.99911 
.99910 
.99909 
•99907 
.99906 
.99905 
•99904 
.99902 
•99901 
.99900 
.99898 
.99897 
•99896 
•99894 
.99893 


•04220 
.04250 
.04279 
.04308 
•04337 


.05970 
-05999 
-06029 
.06058 
•06087 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.04362 
.04391 
.04420 
.04449 
.04478 
.04507 
.04536 
.04565 
.04594 
.04623 


.04366 
.04395 
•04424 
•04454 
•04483 


.06105 
.06134 
•06163 
•06192 
•06221 


•99813 
•99812 
•99810 
•99808 
•99806 


•06116 
•06145 
•06175 
•06204 
-06233 


30 

29 
28 
27 
26 


35 

36 
37 
38 
3L- 


•04512 
•04541 
•04570 
•04599 
•04628 


22.1640 
22^0217 
21.8813 
21^7426 
21.6056 


•06250 
• .06279 
•06308 
•06337 
.06366 


-99804 
-99803 
-99801 
•99799 
-99797 


•06262 1 15 
-06291 ! 15 
-06321 j 15 
-06350 15 
-06379 1 15 


9687 
8945 
8211 
7483 
6762 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.04653 
.04682 
.04711 
.04740 
.04769 


•99892 
•99890 
•99889 
•99888 
•99886 


•04658 
•04687 
•04716 
•04745 
•04774 


21^4704 
21-3369 
21-2049 
21-0747 
20-9460 


-06395 
•06424 
•06453 
•06482 
•06511 


•99795 
•99793 
•99792 
•99790 
•99788 


•06408 
•06437 
•06467 
•06496 
.06525 


15 
15 
15 
15 
15 
15 
15 
15 
15 
14 


6048 
5340 
4638 
3943 
3254 
2571 
1893 
1222 
0557 
9898 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.04798 
.04827 
.04856 
.04885 
.04914 
.04943 
•04972 
.05001 
.05030 
.05059 
.05088 
.05117 
.05146 
.05175 
.05205 


•99885 
•99883 
•99882 
•99881 
=99879 


•04803 
•04833 
04862 
•04891 
•04920 


20-8188 
20-6932 
20-5691 
20^4465 
20.3253 


•06540 
•06569 
•06598 
•06627 
•06656 


-99786 
-99784 
.99782 
.99780 
.99778 


•06554 
•06584 
•06613 
-06642 
-06671 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•99878 
•99876 
.99875 
.99873 
.99872 


•04949 
•04978 
•05007 
•05037 
•05066 


20^2056 
20^0872 
19-9702 
19-8546 
19-7403 


•06685 
•06714 
•06743 
•06773 
.06802 


.99776 
.99774 
.99772 
.99770 
.99768 


•06700 
-06730 
.06759 
.06788 
-06817 


14 
14 
14 
14 
14 


9244 
8596 
7954 
7317 
6685 


10 

9 
8 
7 
6 


55 
66 
57 
58 
59 


.99870 
.99869 
.99867 
.99866 
.99864 
.99863 


•05095 
.05124 
.05153 
.05182 
•05212 


19-6273 
19-5156 
19.4051 
19-2959 
19-1879 


.06831 
.06860 
.06889 
-06918 
-06947 


.99766 
.99764 
.99762 
.99760 
-99758 


-06847 
.06876 
.06905 
.06934 
-06963 


14 
14 
14 
14 
14 
14 


6059 
5438 
4823 
4212 
3607 
3007 


5 
4 
3 
2 
1 


PO 


.05234 


•05241 


19. 0811 


-06976 


-99756 


•06993 





/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. J 


/ 



87° 



684 



86^ 



-NATURAL SINES, COSINES. TANGENTS. AND COTANGENTS. 

4° 5° 



/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 
4 


.06976 
.07005 
.07034 
.07063 
.07092 


•99756 
.99754 
.99752 
.99750 
.99748 


.06993 
.07022 
.07051 
.07080 
.07110 


14.3007 
14.2411 
14.1821 
14.1235 
14.0655 


.08716 
.08745 
.08774 
.08803 
.08831 


-99619 
-99617 
.99614 
.99612 
.99609 


.08749 
•08778 
•08807 
.08837 
.08866 


11-4301 
11-3919 
11-3540 
11-3163 
11-2789 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.07121 
.07150 
.07179 
.07208 
.07237 


.99746 
.99744 
.99742 
.99740 
.99738 


.07139 
.07168 
.07197 
.07227 
.07256 


14.0079 
13.9507 
13.8940 
13.8378 
13.7821 


.08860 
.08889 
.08918 
.08947 
.08976 


.99607 
.99604 
-99602 
.99599 
.99596 


.08895 
.08925 
.08954 
.08983 
.09013 


11-2417 
11-2048 
11.1681 
11.1316 
11-0954 


55 
54 
53 
52 
51 


10 

11 

12 
13 
14 


.07266 
.07295 
.07324 
.07353 
.07382 


.99736 
.99734 
.99731 
.99729 
.99727 


.07285 
.07314 
.07344 
.07373 
.07402 


13.7267 
13.6719 
13.6174 
13.5634 
13.5098 


.09005 
.09034 
.09063 
.09092 
.09121 


.99594 
.99591 
-99588 
.99586 
99583 


-09042 
-09071 
-09101 
-09130 
.09159 


11-0594 
11.0237 
10-9882 
10-9529 
10-9178 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 
30 
21 
22 
23 
24 


.07411 
.07440 
.07469 
.07498 
.07527 


.99725 
.99723 
.99721 
.99719 
.99716 


.07431 
.07461 
.07490 
.07519 
.07548 


13-4566 
13-4039 
13-3515 
13-2996 
13.2480 


•09150 
.09179 
.09208 
.09237 
•09266 


.99580 
.99578 
.99575 
•99572 
•99570 


-09189 
.09218 
.09247 
.09277 
.09306 


10.8829 
10.8483 
10.8139 
10.7797 
10^7457 


45 
44 
43 
42 
41 


.07556 
.07585 
.07614 
.07643 
.07672 


.99714 
.99712 
.99710 
.99708 
.99705 


.07578 
.07607 
.07636 
.07665 
.07695 


13-1969 
13-1461 
13-0958 
13-0458 
12-9962 


-09295 
.09324 
.09353 
.09382 
.09411 


-99567 
•99564 
•99562 
.99559 
•99556 


-09335 
•09365 
.09394 
.09423 
.09453 


10.7119 
10.6783 
10.6450 
10.6118 
10.5789 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.07701 
.07730 
.07759 
.07788 
.07817 


.99703 
.99701 
.99699 
.99696 
.99694 


.07724 
.07753 
.07782 
.07812 
.07841 


12.9469 
12.8981 
12-8496 
12.8014 
12.7536 


-09440 
.09469 
.09498 
.09527 
•09556 


.99553 
.99551 
.99548 
.99545 
.99542 


.09482 
•09511 
•09541 
•09570 
.09600 


10.5462 
10.5136 
10.4813 
10.4491 
10.4172 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.07846 
.07875 
.07904 
.07933 
.07962 


.99692 
.99689 
.99687 
.99685 
.99683 


.07870 
.07899 
.07929 
.07958 
•07987 


12-7062 
12.6591 
12.6124 
12-5660 
12.5199 


.09585 
.09614 
.09642 
.09671 
.09700 


.99540 
.99537 
.99534 
•99531 
•99528 


.09629 
.09658 
•09688 
.09717 
•09746 


10.3854 
10.3538 
10.3224 
10.2913 
10.2602 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.07991 
.08020 
.08049 
.08078 
•08107 


.99680 
.99678 
.99676 
.99673 
•99671 


.08017 
.08046 
.08075 
•08104 
•08134 


12.4742 
12.4288 
12.3838 
12.3390 
12.2946 


.09729 
.09758 
.09787 
.09816 
•09845 


•99526 
.99523 
.99520 
•99517 
99514 


•09776 
•09805 
•09834 
•09864 
.09893 


10.2294 
10.1988 
10.1683 
iO-1381 
10.1080 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


.08136 
.08165 
.08194 
.08223 
.08252 


.99668 
.99666 
.99664 
.99661 
.99659 


.08163 
.08192 
.08221 
.08251 
.08280 


12.2505 
12-2067 
12-1632 
12-1201 
12.0772 


.09874 
.09903 
•09932 
.09961 
.09990 


•99511 
•99508 
.99506 
.99503 
.99500 


•09923 
.09952 
.09981 
.10011 
. 10040 


10-0780 
10-0483 
10. 0187 
9-98931 
9.96007 


20 

19 
18 
17 
16 


.08281 
.08310 
.08339 
.08368 
.08397 


.99657 
.99654 
.99652 
.99649 
.99647 


.08309 
.08339 
.08368 
.08397 
.08427 


12-0346 
11.9923 
11-9504 
11.9087 
11.8673 


.10019 
.10048 
-10077 
.10106 
.10135 


.99497 
.99494 
.99491 
.99488 
.99485 


.10069 
.10099 
.10128 
.10158 
.10187 


9-93101 
9-90211 
9-87338 
9-84482 
9.81641 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 


.08426 
.08455 
•08484 
.08513 
.08542 


.99644 
.99642 
.99639 
.99637 
.99635 


.08456 
.08485 
.08514 
.08544 
.08573 


11.8262 
11.7853 
11.7448 
11.7045 
11.6645 


•10164 
.10192 
.10221 
.10250 
.10279 


•99482 
•99479 
•99476 
.99473 
.99470 


•10216 
.10246 
•10275 
.10305 
•10334 


9-78817 
9-76009 
9-73217 
9-70441 
9.67680 


10 

9 
8 
7 
6 


.08571 
.08600 
.08629 
.08658 
.08687 


.99632 
.99630 
.99627 
.99625 
.99622 


.08602 
.08632 
.08661 
.08690 
.08720 


11.6248 
11.5853 
11.5461 
11.5072 

11.4685 


.10308 
.10337 
-10366 
-10395 
.10424 


.99467 
.99464 
.99461 
.99458 
.99455 


.10363 
.10393 
. 10422 
-10452 
-10481 
•10510 


9.64935 
9.62205 
9^59490 
9.56791 
9. 54106 


5 

4 
3 

2 

1 


60 


.08716 


•99619 


•08749 


11.4301 


.10453 


•99452 


9. 51436 





/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 1 


Cot. Tan. 


/ 



sr 



685 



84° 



TABLE IX.— 


-NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS 

6° 7° 1 


T.^ 


' Sin. 1 


Cos. 


Tan. 


Cot. Sin. I 


Cos. 


Tan. 


Cot. 


t 


^ 




1 
2 
3 
4 


.10453 
.10482 
.10511 
.10540 
.10569 


.99452 
.99449 
.99446 
.99443 
.99440 


•10510 
.10540 
•10569 
•10599 
•10628 


9.51436 
9.48781 
9.46141 
9.43515 
9^40904 


•12187 
-12216 
-12245 
-12274 
-12302 


•99255 
•99251 
•99248 
.99244 
•99240 


.12278 
•12308 
.12338 
•12367 
.12397 
.12426 
•12456 
•12485 
.12515 
.12544 


8.14435 
8.12481 
8.10536 
8.08600 
8.06674 


6C 

5£ 
5i 
5^* 
56 


I 

1 
2 
3 
j 


5 
6 
7 
8 
9 


.10597 
.10626 
.10655 
.10684 
.10713 


.99437 
.99434 
.99431 
.99428 
.99424 


•10657 
•10687 
.10716 
.10746 
.10775 


9-38307 
9.35724 
9.33155 
9.30599 
9.28058 


.12331 
.12360 
•12389 
-12418 

-12447 


•99237 
•99233 
-99230 
•99226 
•99222 
•99219 
•99215 
.99211 
.99208 
.99204 


8.04756 
8.02848 
8.00948 
7.99058 
7.97176 


55 
54 
5S 
52 
5) 

5C 

4£ 
4£ 
47 
46 
4£ 
44 
4£ 
42 

4C 

38 
38 
37 
36 

3£ 
34 
33 
32 
31 
3C 
29 
28 
27 
26 

25 
24 

23 

22 
21 


« 

« 

J 


10 

11 
12 
13 
14 


.10742 
.10771 
.10800 
.10829 
.10858 


.99421 
.99418 
.99415 
.99412 
.99409 


•10805 
.10834 
.10863 
.10893 
.10922 


9.25530 
9.23016 
9.20516 
9.18028 
9.15554 


•12476 

-12504 

-12533 

.12562 

a2191_ 

-12620 

•12649 

•12678 

.12706 

•12735 

.12764 

•12793 

.12822 

.12851 

.12880 


.12574 
.12603 
.12633 
.12662 
.12692 


7.95302 
7-93438 
7.91582 
7.89734 
7.87895 


r 

11 
12 

IS 


15 
16 
17 
18 
19 
20 
21 
22 
23 
24 


.10887 
.10916 
.10945 
.10973 
.11002 

.11031 
.11060 
.11089 
.11118 
.11147 


.99406 
.99402 
.99399 
.99396 
.99393 
.99390 
.99386 
.99383 
.99380 
.99377 


•10952 
.10981 
•11011 
.11040 
.11070 
•11099 
•11128 
•11158 
.11187 
.11217 


9.13093 
9.10646 
9^08211 
9.05789 
9.03379 

9.00983 
8.98598 
8^96227 
8-93867 
8.91520 


•99200 
.99197 
•99193 
•99189 
.99186 
•99182 
•99178 
•99175 
.99171 
•99167 


.12722 
.12751 
.12781 
.12810 
.12840 
.12869 
•12899 
.12929 
.12958 
.12988 

.13017 
•13047 
•13076 
.13106 
.13136 


7.86064 
7.84242 
7.82428 
7.80622 
7-78825 
7.77035 
7.75254 
7.73480 
7.71715 
7.69957 
7.68208 
7.66466 
7.64732 
7.63005 
7.61287 


1 
1 
1 
1 
1 

2 

2 
2 
2 
2_ 


25 
26 
27 
28 
29 


11176 
•11205 
.11234 
.11263 
.11291 


.99374 
.99370 
.99367 
•99364 
.99360 
.99357 
.99354 
•99351 
.99347 
.99344 


.11246 
•11276 
•11305 
•11335 
•11364 
•11394 
•11423 
.11452 
.11482 
.11511 


8. 89185 
8.86862 
8^84551 
8^82252 
8.79964 


.12908 
.12937 
•12966 
.12995 
.13024 


.99163 
99160 
•99156 
.99152 
^i48_ 
.99144 
.99141 
•99137 
•99133 
•99129 


2' 

2 

2 

2 
2 


30 

31 
32 
33 
34 


•11320 
.11349 
.11378 
.11407 
•11436 


8.77689 
8.75425 
8-73172 
8-70931 
8.68701 


.13053 
•13081 
•13110 
.13139 
.13168 


.13165 
.13195 
.13224 
.13254 
.13284 


7.59575 
7.57872 
7.56176 
7.54487 
7.52806 


3 

3 
3 
3 
3 


35 
36 
37 
38 
39 


.11465 '.99341 
.11494 1-99337 
.11523 1.99334 
.11552 j. 99331 
.11580 1.99327 


•11541 
•11570 
.11600 
•11629 
•11659 


8.66482 
8-64275 
8-62078 
8-59893 
8.57718 


•13197 
•13226 
•13254 
•13283 
•13312 


•99125 
•99122 
•99118 
•99114 
•99110 


.13313 
•13343 
•13372 
.13402 
.13432 


7.51132 
7.49465 
7-47806 
7-46154 
7-44509 


3 


40 

41 
42 
43 
44 


•11609 .99324 

.11638 .99320 

11667 .99317 

11696 .99314 

•11725 ^99310 


.11688 
.11718 
.11747 
•11777 
.11806 


8-55555 
8-53402 
8-51259 
8.49128 
8-47007 


•13341 
•13370 
.13399 
•13427 
•13456 


•99106 
•99102 
.99098 
.99094 
•99091 


.13461 
.13491 
.13521 
.13550 
.13580 


7-42871 
7-41240 
7.39616 
7.37999 
7.36389 


20 
J9 
18 
17 
16 




45 
46 
47 
48 
49 


•11754 
•11783 
.11812 
•11840 
.11869 


.99307 
.99303 
.99300 
•99297 
•99293 


•11836 
•11865 
•11895 
•11924 
.11954 


8-44896 
8-42795 
8.40705 
8.38625 
8.36555 


•13485 
.13514 
.13543 
.13572 
•13600 


•99087 
•99083 
•99079 
•99075 
•99071 


.13609 
.13639 
.13669 
.13698 
.13728 


7.34786 
7.33190 
7.31600 
7.30018 
7.28442 


15 
14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 

O 


4 


50 

51 
52 
53 
54 


.11898 
.11927 
•11956 
.11985 
•12014 


.99290 
.99286 
•99283 
•99279 
•99276 


.11983 
.12013 
.12042 
.12072 
.12101 


8.34496 
8.32446 
8.30406 
8.28376 
8.26355 


•13629 
•13658 
•13687 
•13716 
.13744 


•99067 
•99063 
.99059 
.99055 
.99051 


.13758 
.13787 
•13817 
.13846 
.13876 


7.26873 
7.25310 
7.23754 
7.22204 
7-20661 


5 
5 
5 


55 
56 
57 
58 
59 


•12043 
.12071 
.12100 
.12129 
.12158 


.99272 
.99269 
•99265 
•99262 
•99258 


.12131 
.12160 
•12190 
•12219 
•12249 


8.24345 
8.22344 
8.20352 
8-18370 
8-16398 


.13773 
.13802 
.13831 
.13860 
.13889 
•13917 


•99047 
•99043 
99039 
•99035 
.99031 


.13906 
.13935 
•13965 
.13995 
.14024 
.14054 


7-19125 
7.17594 
7.16071 
7.14553 
7.13042 


1 


60 


.12187 


.99255 


•12278 


8.14435 


.99027 


7-11537 


i 




Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


' 


^ 


83** 686 88** 





-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
8° 9"" 



/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. Cot. 


/ 




1 

2 
3 

4 


.13917 
.13946 
.13975 
.14004 
.14033 


•99027 
•99023 
•99019 
•99015 
.99011 


•14054 
.14084 
.14113 
.14143 
.14173 


7^11537 
7.10038 
7.08546 
7.07059 
7.05579 


.15643 
.15672 
.15701 
.15730 
.15758 


-98769 
.98764 
.98760 
.98755 
-98751 


•15838 
•15868 
.15898 
•15928 
•15958 


6.31375 
6.30189 
6.29007 
6.27829 
6.26655 


60 

59 
58 

57 
56 


5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 


.14061 
.14090 
.14119 
•14148 
.14177 


.99006 
.99002 
.98998 
.98994 
.98990 


.14202 
•14232 
.14262 
.14291 
.14321 


7.04105 
7.02637 
7.01174 
6.99718 
6.98268 


.15787 
.15816 
.15845 
.15873 
15902 


-98746 
.98741 
.98737 
.98732 
.98728 


•15988 
•16017 
.16047 
.16077 
.16107 


6.25486 
6.24321 
6.23160 
6.22003 
6.20851 


55 
54 
53 
52 
51 


.14205 
.14234 
.14263 
.14292 
.14320 


.98986 
.98982 
.98978 
.98973 
.98969 


.14351 
•14381 
•14410 
. 14440 
• 14470 


6- 96823 
6^95385 
6-93952 
6-92525 
6^91104 


.15931 
•15959 
.15988 
.16017 
.16046 


.98723 
•98718 
•98714 
.98709 
•98704 


.16137 
.16167 
.16196 
.16226 
.16256 


6.19703 
6.18559 
6.17419 
6.16283 
6.15151 


50 

49 
48 
47 
46 


.14349 
.14378 
.14407 
.14436 
14464 


.98965 
.98961 
.98957 
.98953 
.98948 


•14499 
•14529 
•14559 
•14588 
•14618 


6-89688 
6-88278 
6-86874 
6^85475 
6 •84082 


.16074 
.16103 
-16132 
.16160 
.16189 


•98700 
•98695 
•98690 
•98686 
•98681 


.16286 
.16316 
.16346 
.16376 
•16405 


6.14023 
6.12899 
6.11779 
6.10664 
6.09552 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


■14493 
.14522 
•14551 
.14580 
.14608 


.98944 
.98940 
•98936 
.98931 
•98927 


•14648 
.14678 
.14707 
.14737 
.14767 


6^82694 
6-81312 
6-79936 
6.78564 
6.77199 


-16218 
.16246 
.16275 
.16304 
.16333 


•98676 
•98671 
.98667 
.98662 
•98657 


•16435 
•16465 
•16495 
•16525 
.16555 

•16585 
.16615 
.16645 
.16674 
.16704 


6.08444 
6.07340 
6.06240 
6.05143 
6.04051 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.14637 
.14666 
.14695 
.14723 
.14752 


•98923 
•98919 
•98914 
•98910 
•98906 


.14796 
.14826 
.14856 
.14886 
.14915 


6. 75838 
6-74483 
6-73133 
6.71789 
6.70450 


.16361 
.16390 
.16419 
.16447 
.16476 


•98652 
•98648 
•98643 
•98638 
•98633 


6.02962 
6.01878 
6-00797 
5.99720 
5.98646 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.14781 
.14810 
.14838 
.14867 
.14896 


•98902 
•98897 
•98893 
.98889 
.98884 


.14945 
.14975 
.15005 
.15034 
.15064 


6-69116 
6-67787 
6-66463 
6-65144 
6.63831 


.16505 
.16533 
-16562 
-16591 
-16620 


.98629 
.98624 
.98619 
.98614 
.98609 


.16734 
.16764 
•16794 
•16824 
•16854 


5.97576 
5-96510 
5-95448 
5.94390 
5.93335 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.14925 
.14954 
.14982 
.15011 
.15040 


.98880 
.98876 
.98871 
.98867 
.98863 


.15094 
.15124 
.15153 
.15183 
.15213 


6.62523 
6.61219 
6.59921 
6.58627 
6.57339 


.16648 
.16677 
.16706 
.16734 
.16763 


•98604 
.98600 
.98595 
98590 
.98585 


.16884 
.16914 
.16944 
.16974 
.17004 


5.92283 
5.91236 
5.90191 
5.89151 
5-88114 


25 
24 
23 
22 
21 


40 

11 
12 
13 
14 


.15069 
.15097 
.15126 
.15155 
.15184 


.98858 
.98854 
.98849 
.98845 
98841 


.15243 
.15272 
.15302 
.15332 
.15362 


6.56055 
6.54777 
6.53503 
6.52234 
6.50970 


.16792 
.16820 
.16849 
.16878 
.16906 


.98580 
.98575 
.98570 
.98565 
•98561 


.17033 
.17063 
.17093 
.17123 
.17153 


5-87080 
5-86051 
5-85024 
5.84001 
5.82982 


30 

19 
18 
17 
16 


15 
16 
17 
18 
19 
50 
51 
52 
53 
54 

55 
56 
)7 


.15212 
.15241 
.15270 
.15299 
•15327 


•98836 
•98832 
•98827 
.98823 
.98818 


.15391 
.15421 
.15451 
.15481 
.15511 


6.49710 
6.48456 
6.47206 
6.45961 
6.44720 


•16935 
.16964 
-16992 
-17021 
•17050 


•98556 
.98551 
.98546 
.98541 
•98536 


.17183 
•17213 
.17243 
.17273 
.17303 


5.81966 
5.80953 
5.79944 
5.78938 
5.77936 


15 
14 
13 
12 
11 


.15356 
.15385 
.15414 
• 15442 
•15471 


•98814 
•98809 
•98805 
•98800 
.98796 


.15540 
.15570 
.15600 
.15630 
.15660 


6.43484 
6.42253 
6.41026 
6.39804 
6-38587 


•17078 
•17107 
-17136 
.17164 
•17193 


•98531 
•98526 
•98521 
•98516 
•98511 


.17333 
.17363 
.17393 
.17423 
.17453 


5.76937 
5.75941 
5.74949 
5.73960 
5.72974 
5.71992 
5-71013 
5-70037 
5-69064 
5-68094 


10 

9 
8 

7 
6 


•15500 
• 15529 
•15557 
•15586 
.15615 


.98791 
•98787 
•98782 
•98778 
•98773 


.15689 
.15719 
.15749 
.15779 
•15809 


6-37374 
6.36165 
6-34961 
6.33761 
6-32566 


.17222 
-17250 
-17279 
-17308 
.17336 


.98506 
.98501 
.98496 
•98491 
•98486 


.17483 
.17513 
.17543 
•17573 
.17603 


5 

4 
3 
2 

I 


30 


•15643 


.98769 


•15838 


6-31375 


-17365 


•98481 


.17633 


5-67128 





f 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


' 


81** 687 80** 



TABLE IX.--NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
10° 11° 



» 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


f 




1 

2 
3 
4 


.17365 
.17393 
17422 
.17451 
.17479 


.98481 
.98476 
.98471 
.98466 
.98461 


.17633 
.17663 
.17693 
.17723 
.17753 


5.67128 
5-66165 
5-65205 
5 •64248 
5 •63295 


-19081 
.19109 
-19138 
.19167 
-19195 


98163 
■98157 
•98152 
•98146 
•98140 


.19438 
.19468 
•19498 
•19529 
.19559 


5 
5 
5 
5 
5 


14455 
13658 
12862 
12069 
11279 


60 

59 
58 
57 
56 


5 
6 
7 
8 

9 


.17508 
.17537 
.17565 
.17594 
.17623 


.98455 
.98450 
.98445 
. 98440 
.98435 


.17783 
.17813 
.17843 
.17873 
.17903 


5-62344 
5^61397 
5-60452 
5-59511 
5-58573 


•19224 
.19252 
-19281 
.19309 
-19338 


•98135 
•98129 
•98124 
•98118 
98112 


.19589 
.19619 
•19649 
.19680 
•19710 


5 
5 
5 
5 
5 


10490 
09704 
08921 
08139 
07360 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.17651 
.17680 
.17708 
.17737 
.17766 


.98430 
.98425 
.98420 
•98414 
.98409 


17933 
.17963 
.17993 
* 18023 
.18053 


5-57638 
5.56706 
5.55777 
5.54851 
5.53927 


.19366 
.19395 
.19423 
-19452 
-19481 


•98107 
■98101 
■98096 
•98090 
.98084 


.19740 
.19770 
.19801 
•19831 
.19861 


5 
5 
5 
5 
5 


06584 
05809 
05037 
04267 
03499 


50 

49 
48 
47 

46 


15 
16 
17 
18 
19- 


.17794 
.17823 
.17852 
.17880 
.17909. 
17937 
.17966 
.17995 
.18023 
.18052 

.18081 
.18109 
.18138 
.18166 
.18195 


.98404 
•98399 
.98394 
.98389 
.98383 


.18083 
.18113 
.18143 
.18173 
•18203 


5.53007 
5.52090 
5.51176 
5.50264 
5.49356 


.19509 
.19538 
-19566 
-19595 
•19623 


•98079 
•98073 
•98067 
.98061 
.98056 


.19891 
.19921 
.19952 
.19982 
•20012 


5 
5 
5 
5 
4 


02734 
01971 
01210 
00451 
99695 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


.98378 
•98373 
.98368 
.98362 
•98357 


•18233 
.18263 
.18293 
•18323 
•18353 


5.48451 
5.47548 
5-46648 
5-45751 
5 •44857 


•19652 
•19680 
•19709 
-19737 
-19766 


.98050 
.98044 
.98039 
.98033 
.98027 


•20042 
•20073 
-20103 
.20133 
.20164 


4 
4 
4 
4 
4 


98940 
98188 
97438 
96690 
95945 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•98352 
.98347 
.98341 
.98336 
.98331 


.18384 
.18414 
. 18444 
.18474 
.18504 


5-43966 
5.43077 
5-42192 
5-41309 
5-40429 


-19794 
.19823 
•19851 
19880 
•19908 


.98021 
.98016 
•98010 
•98004 
•97998 


•20194 
•20224 
-20254 
-20285 
-20315 


4 
4 
4 
4 
4 


95201 
94460 
93721 
92984 
G2249 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.18224 
.18252 
.18281 
.18309 
.18338 


•98325 
•98320 
•98315 
•98310 
•98304 


.18534 
•18564 
•18594 
•18624 
•18654 


5-39552 
5-38677 
5-37805 
5-36936 
5-36070 


19937 
•19965 
•19994 
■20022 

20051 


•97992 
•97987 
•97981 
•97975 
.97969 


-20345 
-20376 
-20406 
-20436 
-20466 


4 
4 
4 
4 
4 


91516 
90785 
90056 
89330 
88605 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.18367 
.18395 
.18424 
.18452 
.18481 


•98299 
•98294 
•98288 
•982«3 
•98277 


•18634 
•18714 
•18745 
.18775 
.18805 


5-35206 
5^34345 
5-33487 
5-32631 
5. 31778 


•20079 
-20108 
-20136 
-20165 
•20193 


•97963 
•97958 
.97952 
.97946 
.97940 


-20497 
.20527 
-20557 
-20588 
•20618 


4 
4 
4 
4 
4 


87882 
87162 
86444 
85727 
85013 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


• 18509 
•18538 
.18567 
.18595 
.18624 


•98272 
•98267 
•98261 
•98256 
.98250 


.18835 
.18865 
.18895 
•18925 
.18955 


5-30928 
5-30080 
5-29235 
5.28393 
5-27553 


•20222 
•20250 
•20279 
•20307 
•20336 


.97934 
.97928 
.97922 
.97916 
•97910 


•20648 
•20679 
.20709 
.20739 
•20770 


4 
4 
4 
4 
4 


84300 
83590 
82882 
82175 
81471 


30 

19 
18 
17 
18 


45 
46 
47 
48 
49 


.18652 
.18681 
•18710 
18738 
•18767 


.98245 
.98240 
.98234 
.98229 
.98223 


.18986 
.19016 
.19046 
.19076 
.19106 


5.26715 
5-25880 
5-25048 
5.24218 
5-23391 


•20364 
•20393 
•20421 
•20450 
•20478 


•97905 
-97899 
•97893 
•97887 
•97881 


•20800 
•20830 
•20861 
.20891 
•20921 


4 
4 
4 
4 
4 


80769 
80068 
79370 
78673 
77978 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


18795 
.18824 
.18852 
.18881 
.18910 


•98218 
.98212 
•98207 
.98201 
.98196 


.19136 
•19166 
•19197 
.19227 
.19257 


5-22566 
5-21744 
5-20925 
5.20107 
5-19293 


•20507 
•20535 
•20563 
•20592 
•20620 


•97875 
•97869 
•97863 
•97857 
•97851 


•20952 
•20982 
•21013 
•21043 
•21073 


4 
4 
4 
4 
4 


77286 
76595 
75906 
75219 
74534 


10 

9 

a 

7 
6 


55 
56 
57 
58 
59 


.18938 
•18967 
.18995 
.19024 
.19052 


.98190 
.98185 
.98179 
.98174 
.98168 


.19287 
•19317 
•19347 
•19378 
•19408 


5-18480 
5.17671 
5.16863 
5-16058 
5-15256 


•20649 
•20677 
•20706 
•20734 
•20763 


•97845 
•97839 
•97833 
•97827 
•97821 


•21104 
•21134 
•21164 
•21195 
•21225 


4 

4 
4 
4 
4 


73851 
73170 
72490 
71813 
71137 


5 

4 
3 

2 

1 


60_ 


.19081 


.98163 


19438 


5-14455 


•20791 


•97815 


•21256 


4 


70463 





'-T- 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


"^ 



79° 



688 



78* 



-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
12° 13° 



/ 


Sin. 


Cos. 


Tan. Cot. | 


Sin. 


Cos. 


Tan. 


Cot. 


' 




1 

2 
3 
4 


20791 
20820 
.20848 
.20877 
•20905 


•97815 
•97809 
•97803 
•97797 
•97791 


•21256 
.21286 
.21316 
.21347 
.21377 
.21408 
.21438 
.21469 
.21499 
•21529 


4 
4 
4 
4 
4 


70463 
69791 
69121 
68452 
67786 


•22495 
.22523 
.22552 
•22580 
•22608 


.97437 
.97430 
•97424 
•97417 
•97411 


•23087 
•23117 
.23148 
•23179 
•23209 


4 
4 
4 

t 


33148 
32573 
32001 
31430 
30860 


60 

59 
58 
57 
56 


5 

6 
7 
8 
9 


.20933 1 
• 20962 
20990 i 
.21019 
•21047 > 


•97784 
97778 
•97772 
•97766 
•97760 


4 
4 
4 
4 
4 


67121 
66458 
65797 
65138 
64480 


•22637 
•22665 
-22693 
•22722 
•22750 


•97404 
.97398 
•97391 
•97384 
•97378 


•23240 
•23271 
.23301 
•23332 
•23363 


4 
4 
4 
4 
4 


30291 
29724 
29159 
28595 
28032 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.21076 i 

.21104 1 

.21132 1 

.21161 

.21189 

.21218 

.21246 

.21275 

• 21303 

•21331 


•97754 
•97748 
•97742 
•97735 
•97729 


•21560 
•21590 
21621 
.21651 
•21682 


4 
4 
4 
4 
4 


63825 
63171 
62518 
61868 
61219 


•22778 
•22807 
•22835 
.22863 
•22892 


•97371 
•97365 
.97358 
.97351 
•97345 


23393 
•23424 
•23455 
.23485 
•23516 


4 
4 
4 
4 
4 


27471 
26911 
26352 
25795 
25239 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.97723 
.97717 
•97711 
97705 
•97698 


•21712 
.21743 
.21773 
.21804 
•21834 


4 
4 
4 
4 
4 


60572 
59927 
59283 
58641 
58001 


.22920 
.22948 
•22977 
.23005 
.23033 


•97338 
•97331 
•97325 
.97318 
.97311 


-23547 
.23578 
23608 
•23639 
•23670 


4 
4 
4 
4 
4 


24685 
24132 
23580 
23030 
22481 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


•21360 
•21388 
•21417 
• 21445 
•21474 


•97692 
•97686 
•97680 
•97673 
•97667 


•21864 
.21895 
•21925 
•21956 
.21986 


t 

4 
4 
4 


57363 
56726 
56091 
55458 
54826 


.23062 
•23090 
•23118 
•23146 
•23175 


.97304 
97298 
•97291 
•97284 
97278 


.23700 
.23731 
.23762 
.23793 
.23823 


4 
4 
4 
4 
4 


21933 
21387 
20842 
20298 
19756 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•21502 
.21530 
•21559 
•21587 
.21616 1 


.97661 
•97655 
•97648 
•97642 
•97636 


•22017 
.22047 
.22078 
.22108 
.22139 


4 
4 
4 
4 
4 


54196 
53568 
52941 
52316 
51693 


•23203 
•23231 
.23260 
.23288 
•23316 


.97271 
•97264 
•97257 
.97251 
•97244 


•23854 
•23885 
•23916 
•23946 
.23977 


4 
4 
4 
4 
4 


19215 
18675 
18137 
17600 
17064 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.21644 
• 21672 
21701 
•21729 
•21758 j 


•97630 
•97623 
•97617 
•97611 
.97604 


•22169 
•22200 
•22231 
•22261 
.22292 


4 
4 
4 
4 
4 


51071 
50451 
49832 
49215 
48600 


•23345 
.23373 
•23401 
.23429 
23458 


.97237 
.97230 
.97223 
.97217 
•97210 


.24008 
.24039 
.24069 
.24100 
.24131 
.24162 
.24193 
.24223 
.24254 
•24285 


4 
4 
4 
4 

4 
4 
4 
4 
4 


16530 
15997 
15465 
14934 
14405 
13877 
13350 
12825 
12301 
11778 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


•21786 
•21814 ! 

• 21843 
•21871 ! 

• 21899 


•97598 
.97592 
•97585 
•97579 
•97573 


•22322 
•22353 
•22383 
•22414 
. 22444 


4 
4 
4 
4 
4 


47986 
47374 
46764 
46155 
45548 


23486 
.23514 
.23542 
.23571 
.23599 


•97203 
.97196 
•97189 
•97182 
•97176 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 


.21928 
.21956 
.21985 
.22013 
.22041 


•97566 
•97560 
•97553 
•97547 
•97541 


•22475 
•22505 
.22536 
•22567 
•22597 


4 
4 
4 
4 
4 


44942 
44338 
43735 
43134 
42534 


•23627 
•23656 
.23684 
•23712 
.23740 


.97169 
.97162 
•97155 
.97148 
.97141 


•24316 
.24347 
.24377 
.24408 
.24439 


4 
4 
4 
4 
4 


11256 
10736 
10216 
09699 
09182 


30 

19 
18 
17 
16 


•22070 
.22098 
.22126 
•22155 
.22183 


•97534 
•97528 
.97521 
.97515 
•97508 


.22628 
.22658 
•22689 
•22719 
.22750 


4 
4 
4 
4 
4 


41936 
41340 
40745 
40152 
39560 


.23769 
•23797 
•23825 
•23853 
•23882 


.97134 
•97127 
•97120 
•97113 
•97106 


.24470 
.24501 
.24532 
.24562 
.24593 


4 
4 
4 
4 
4 


08666 
08152 
07639 
07127 
06616 


15 
14 
13 
12 
11 


.22212 
.22240 
.22268 
•22297 
•22325 


•97502 
.97496 
.97489 
.97483 
•97476 


•22781 
•22811 
•22842 
•22872 
.22903 


4 
4 
4 
4 
4 


38969 
38381 
37793 
37207 
36623 


•23910 
•23938 
.23966 
.23995 
•24023 


•97100 
.97093 
.97086 
.97079 
.97072 


.24624 
.24655 
.24686 
.24717 
.24747 


4 
4 
4 
4 
4 


06107 
05599 
05092 
04586 
04081 


10 

9 
8 
7 
6 


55 
56 
157 
158 
!59 


.22353 
•22382 
•22410 
•22438 
•22467 


•97470 
•97463 
.97457 
.97450 
.97444 


•22934 
.22964 
•22995 
.23026 
.23056 


4 
4 
4 
4 
4 


36040 
35459 
34879 
34300 
33723 


•24051 
•24079 
•24108 
•24136 
•24164 


.97065 
.97058 
.97051 
.97044 
.97037 


•24778 
•24809 
•24840 
•24871 
•24902 


4 
4 
4 
4 
4 


03578 
03076 
02574 
02074 
01576 


5 

4 
3 

2 

1 


60 


.22495 


•97437 1.23087 


4 


33148 


•24192 


•97030 


•24933 


4 


01078 





' 


Cos. 


Sin. 1 Cot. 


Tan. 1 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



77^ 



689 



76^ 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS, 
14° 15° 



' 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


r\)t. 


' 




1 

2 
3 
4 


■24192 
.24220 
.24249 
.24277 
.24305 


97030 
•97023 
•97015 
•97008 
•97001 


•24933 
■24964 
■24995 
•25026 
■25056 
■25087 
.25118 
.25149 
.25180 
•25211 


4^01078 
4^00582 
4 00086 
3 99592 
3 99099 


■25882 

.25910 
■25938 
■25966 
■25994 


■96593 
■96585 
•96578 
•96570 
•96562 


•26795 
■26826 
■26857 
■26888 
■26920 
■26951 
26982 
■27013 
■27044 
•27076 


3 
3 
3 

3 
3 


73205 
72771 
72338 
71907 
71476 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.24333 
.24362 
.24390 
.24418 
. 24446 


•96994 
96987 

•96980 
96973 
96966 


3. 98607 
3-98117 
3 97627 
397139 
3. 96651 


•26022 
•26050 
■26079 
■26107 
•26135 


•96555 
•96547 
•96540 
•96532 
•96524 


3 
3 
3 
3 
3 


71046 
•70616 
•70188 
-69761 
•69335 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.24474 
.24503 

24531 
.24559 
.24587 

24615 
. 24644 
.24672 

24700 

24728 


96959 
.96952 
96945 
96937 
•96930 


.25242 
25273 
•25304 
•25335 
■25366 


1 396165 
395680 
395196 
3. 94713 
3 94232 


■26163 
■26191 
■26219 
■26247 
■26275 


■96517 
■96509 
■96502 
■96494 
•96486 


•27107 
•27138 
■27169 
■27201 
■27232 


3 
3 
3 
3 
3 


68909 
■68485 
■68061 
■67638 
■67217 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


•96923 
96916 
96909 
96902 
96894 


■25397 
■25428 
■25459 
•25490 
•25521 


393751 
393271 
392793 
392316 
3 91839 


■26303 
■26331 
■26359 
•26387 
■26415 


•96479 
■96471 
■96463 
■96456 
-96448 


■27263 
•27294 
•27326 
•27357 
.27388 


3 
3 
3 
3 
3 


■66796 
■66376 
■65957 
•65538 
■65121 


45 
44 
43 
42 

41 


20 

21 
22 
23 
24 


24756 

24784 

24813 

•24841 

.24869 


96887 

96880 

96873 

96866 

96858 

96851 ' 

•96844 1 

•96837 

•96829 

•96822 


.25552 391364 
.25583 3 .90890 
•25614 : 390417 
•25645 3 89945 
.25676 ' 3. 89474 


■26443 
26471 
26500 
26528 
26556 


■96440 
■96433 
•96425 
•96417 
•96410 
•96402 
•96394 
96386 
96379 
•96371 
•96363 
96355 
96347 
96340 
•96332 
•96324 
96316 
•96308 
96301 
•96293 


•27419 

•27451 

■27482 

■27513 

•27545 

•27576 

■27607 

■27638 

■27670 

•2770X 

•27732 

•27764 

•27795 

■27826 

■27858 


3 
3 
3 
3 
3 


■64705 
■64289 
■63874 
■63461 
■ 63048 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


24897 

.24925 

24954 

24982 

25010 

.25038 

25066 

25094 

.25122 

.25151 

.25179 
25207 : 

.25235 ; 

.25263 i 

.25291 1 
25320 i 
25348 
25376 
25404 

•25432 


■25707 
25738 
■25769 
■25800 
■25831 


3 89004 
388536 
3 88068 
3 -87601 
387136 


■26584 
■26612 
■26640 
26668 
26696 


3 
3 
3 
3 
3 


•62636 
•62224 
•61814 
•61405 
60996 


35 

34 
33 
32 
31 


30 

31 
32 
33 
34 


•96815 25862 
96807 25893 
•96800 .25924 
96793 .25955 
96786 .25986 
■96778 .26017 
96771 i. 26048 
96764 '■ 26079 
■96756 ,■26110 
■96749 .26141 
■96742 .26172 
•96734 26203 
•96727 .26235 
•96719 .26266 
•96712 .26297 


386671 
386208 
3 85745 
3 85284 
3 84824 


26724 
•26752 
•26780 
•26808 
■26836 

26864 
•26892 

26920 
•26948 

26976 


3 
3 
3 
3 
3 


60588 
60181 
59775 
59370 
58966 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


3 84364 
3 83906 
3 83449 
3. 82992 
3 82537 


■27889 
•27921 
•27952 
•27983 
•28015 


3 
3 
3 
3 
3 


58562 
58160 
57758 
57357 
56957 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


3. 82083 
3 81630 
381177 
3 80726 
3 80276 


27004 
27032 

•27060 
27088 

■27116 


■96285 
•96277 
•96269 
■96261 
96253 


•28046 

•28077 

■28109 

•28140 

•28172 

•28203 

•28234 

•28266 

•28297 

•28329^ 

■28360 

■28391 

•28423 

•28454 

■28486 


3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 


56557 
56159 
55761 
55364 
54968 
54573 
54179 
53785 
53393 
53001 
52609 
52219 
51829 
51441 
51053 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•25460 
•25488 
•25516 
•25545 
.25573 


.96705 .26328 
•96697 .26359 
•96690 .26390 
96682 ^26421 
•96675 i. 26452 


3-79827 
3 79378 
3 ■ 78931 
3 78485 
3 ■ 78040 


■27144 
■27172 
•27200 
•27228 
■27256 


■96246 
•96238 
■96230 
•96222 
•96214 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.25601 
•25629 
•25657 
•25685 
•25713 


■96667 ' 
.96660 
•96653 , 
.96645 
•96638 


•26483 
•26515 
•26546 
•26577 
•26608 


377595 
3^77152 
3-76709 
3.76268 
3 .75828 


■27284 
■27312 
■27340 
■27368 
■27396 
■ 27424 
■27452 
■27480 
•27508 
•27536 


•96206 
•96198 
•96190 
.96182 
•96174 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•25741 
.25769 
.25798 
•25826 
•25854 


•96630 
.96623 
.96615 i 
.96608 
.96600 


.26639 
.26670 
.26701 
.26733 
•26764 
•26795 


3 75388 
3-74950 
3-74512 
3^74075 
3^73640 


.96166 
.96158 
.96150 
.96142 
.96134 


■28517 
■28549 
■28580 
•28612 
.28643 


3 

3. 

3. 

3. 

3. 


50666 
50279 
49894 
49509 
49125 


5 
4 
3 
2 
1 


60 


.25882 


.96593 


3 ■73205 


■27564 


■96126 


■28675 


-3^ 


48741 





' 


Cos. 


Sin. 


Cot. 


Tan. 


Co.>^. 


Sin. Cot. 1 


Tan. 1 


' 



75° 



690 



740 



TABLE IX.~NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
16° 17° 



' 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 1 Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 
4 


.27564 
.27592 
.27620 
.27648 
.27676 


•96126 
.96118 
.96110 
.96102 
.96094 


•28675 
•28706 
28738 
•28769 
•28800 


3 48741 
3-48359 
347977 
347596 
3.47216 


-29237 
-29265 
-29293 
-29321 
.29348 


|^95630 
.•95622 
1.95613 
1.95605 
i.95596 


•30573 
•30605 
•30637 
•30669 
•30700 


3-27085 
3-26745 
3-26406 
3^26067 
3^25729 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.27704 
.27731 
.27759 
.27787 
.27815 


.96086 
.96078 
.96070 
•96062 
.96054 


•28832 
•28864 
•28895 
•28927 
.28958 


3-46837 
3-46458 
3 46080 
3.45703 
3.45327 


-29376 
-29404 
-29432 
.29460 
•29487 


•95588 
•95579 
.95571 
.95562 
•95554 


•30732 
•30764 
•30796 
•30828 
•30860 


3^25392 
3-25055 
3-24719 
3-24383 
3 - 24049 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.27843 
.27871 
.27899 
.27927 
.27955 
.27983 
.28011 
.28039 
.28067 
.28095 

.28123 
• 28150 
•28178 
•28206 
•28234 


.96046 
.96037 
.96029 
.96021 
•96013 


•28990 
•29021 
.29053 
•29084 
•29116 


3.44951 
3.44576 
3.44202 
3.43829 
3-43456 


.29515 
.29543 
.29571 
•29599 
-29626 


.95545 
.95536 
.95528 
.95519 
•95511 


.30891 
.30923 
•30955 
•30987 
•31019 


3-23714 
3.23381 
3.23048 
3.22715 
3.22384 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 

20 

21 
22 
23 
24 


•96005 
•95997 
•95989 
•95981 
.95972 

.95964 
.95956 
.95948 
.95940 
•95931 


•29147 
.29179 
.29210 
.29242 
.29274 
.29305 
•29337 
•29368 
•29400 
.29432 


3.43084 
3-42713 
3.42343 
3-41973 
! 3.41604 

3.41236 
3.40869 
3.40502 
3.40136 
3.39771 


•29654 
.29682 
.29710 
.29737 
.29765 

.29793 
.29821 
.29849 
.29876 
•29904 


.95502 
.95493 
.95485 
.95476 
.95467 

.95459 
.95450 
.95441 
•95433 
•95424 


•31051 
.31083 
•31115 
•31147 
•31178 

•31210 
•31242 
•31274 
•31306 
•31338 


3.22053 
3 21722 
3-21392 
3-21063 
3-20734 

3-20406 
3.20079 
3.19752 
3^19426 
3-19100 


45 
44 
43 
42 
41 
40 
39 
38 
37 
36 


25 
26 
27 
28 
29 


• 28262 
•28290 
•28318 
•28346 
•28374 


.95923 
•95915 
.95907 
•95898 
.95890 


•29463 
•29495 
•29526 
.29558 
.29590 


3.39406 
3.39042 
3.38679 
3^38317 
3-37955 


•29932 
•29960 
•29987 
•30015 
•30043 


•95415 
•95407 
•95398 
•95389 
•95380 


•31370 
-31402 
-31434 
-31466 
.31498 


3-18775 
3^18451 
3^18127 
3-17804 
3-1748] 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


•28402 
•28429 
28457 
•28485 
•28513 


•95882 
•95874 
.95865 
•95857 
.95849 


•29621 
.29653 
.29685 
•29716 
.29748 


3-37594 
3-37234 
336875 
3-36516 
3-36158 


•30071 
•30098 
-30126 
•30154 
•30182 


• 95372 i. 31530 
•95363 (•31562 
.95354 1.31594 
.95345 .31626 
.95337 1-31658 


3-17159 
3-16838 
3^16517 
3^16197 
3^15877 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 
40 
41 
42 
43 
44 


.28541 
•28569 
28597 
•28625 
•28652 
.28680 
.28708 
•28736 
.28764 
.28792 


.95841 
.95832 
.95824 
.95816 
.95807 
.95799 
.95791 
•95782 
•95774 
•95766 


.29780 
.29811 
•29843 
.29875 
.29906 

.29938 
.29970 
•30001 
•30033 
•30065 


3-35800 
3-35443 
3-35087 
3-34732 
3-34377 
3-34023 
3-33670 
3-33317 
3-32965 
3-32614 


•30209 
.30237 
•30265 
•30292 
•30320 
•30348 
.30376 
-30403 
-30431 
.30459 


.95328 
•95319 
-95310 
-95301 
95293 
•95284 
•95275 
•95266 
•95257 
•95248 


-31690 
.31722 
•31754 
•31786 
•31818 
•31850 
.31882 
-31914 
.31946 
.31978 


3-15558 
3-15240 
3-14922 
3-14605 
3-14288 
3-13972 
3-13656 
3 • 13341 
3-13027 
3-12713 


25 
24 
23 
22 
21 

20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.28820 
.28847 
.28875 
•28903 
•28931 
.28959 
.28987 
.29015 
.29042 
.29070 


•95757 
.95749 
.&5740 
•95732 
.95724 


.30097 
•30128 
•30160 
•30192 
•30224 


3.32264 
3.31914 
3.31565 
3.31216 
330868 
3^30521 
3^30174 
3^29829 
3 • 29483 
3-29139 


.30486 
•30514 
•30542 
•30570 
•30597 


•95240 
.95231 
•95222 
•95213 
•95204 


•32010 
•32042 
•32074 
•32106 
-32139 


3-12400 
3-12087 
3^11775 
3^11464 
3-11153 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•95715 
•95707 
•95698 
•95690 
.95681 

•95673 
.95664 
.95656 
•95647 
95639 


•30255 
.30287 
.30319 
•30351 
•30382 


•30625 
.30653 
.30680 
.30708 
•30736 


•9S195 
.95186 
•95177 
•95168 
•95159 


-32171 
.32203 
•32235 
.32267 
.32299 


3-10842 
3^10532 
3^10223 
3.09914 
3.09606 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.29098 
.29126 
•29154 
•29182 
•29209 


•30414 
•30446 
.30478 
.30509 
•30541 


3-28795 
3.28452 
3.28109 
3.27767 
3-27426 


.30763 
.30791 
•30819 
.30846 
.30874 


.95150 
•95142 
.95133 
•95124 
•95115 


•32331 
•32363 
•32396 
•32428 
•32460 


3.09298 
3^08991 
3. 08685 
3. 08379 
3. 08073 


5 

4 
3 

2 

1 


go. 


•29237 


•95630 


•30573 


3.27085 


•30902 


•95106 


•32492 


3 •07^68 







Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. Tan. | 


/ 



73° 



691 



TAB L 10 IX. 



-NATURAL SINKS, (^OSINIvS, TANGENTS. AND (X)TAN(iKNTS. 
18° 19° 



/ 


Sin. 

.30902 
.30929 
.30957 
.30985 
.31012_ 

31040 
.31068 
31095 
31123 
•31151, 
.31178 
31206 
31233 
31261 
31289 
31316' 
31344 
31372 
31399 
31427 


Cos. 


Tan. 


Cot. 1 


Sin. 1 Cos. 1 


Tan. 


Cot. 


/ 


o 

1 

2 
3 
4 


.95106 
.95097 
.95088 
.95079 
.95070 
.95061 
.95052 
.95043 
.95033 
.95024 

.95015 
.95006 
94997 
.94988 
.94979 
.94970 
.94961 
.94952 
.94943 
.94933 


32492 
32524 
32556 
32588 
•32621_ 
•32653 
32685 
•32717 
•32749 
.32782_ 

32814 
32846 
32878 
.32911 
.32943_ 

32975 
33007 
33040 

33072 
33104 


3 
3 
3 
3 
3 


07768 
07464 
07160 
06857 
06554 


32557 1 

32584 

32612 

32639 

.32667_ 

•32694 

•32722 

32749 

32777 

.32804_ 

32832 

32859 

32887 

•32914 

.32942_ 

•32969 

32997 

33024 

.33051 

33079 


•94552 
•94542 

94533 
•94523 

94514 


•34433 
•34465 
•34498 
•34530 
•34563 
•34596 
.34628 
.34661 
•34693 
.34726. 

•34758 
•34791 
•34824 
•34856 
.34889 
.34922^ 
34954 
34987 
.35020 
.35052 


2 
2 
2 

2 
2 


90421 
90147 
89873 
89600 
89327 


(iO 

59 
58 
57 
56 


5 
6 
7 
8 
9 


3 
3 
3 
3 
3 


06252 
05950 
05649 
05349 
05049 


94504 
•94495 
•94485 
•94476 
• 94466 _ 
•94457 
•94447 
•94438 
•94428 
•94418 
.94409^ 
.94399 
94390 
94380 
.94370 


2 
2 
2 
2 
2 
2 
2 
2 

2 
2 
2 
2 
2 
2 
2 


89055 
88783 
88511 
88240 
87970 
87700 
.87430 
87161 
86892 
•86624 
.86356 
•86089 
•85822 
•85555 
85289 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


3 
3 
3 
3 
3 


04749 
04450 
04152 
03854 
03556 


r>o 

49 
48 
47 
4o 


15 
16 
17 
18 
19 


3 
3 
3 
3 
3 


03260 
02963 
02667 
02372 
02077 


45 
44 
43 
42 
41 


40 

21 
22 
23 
24 


31454 
31482 
31510 
31537 
J156^ 

31593 
31620 
31648 
31675 
31703 
31730 
31758 
31786 
31813 
^1841 

31868 
31896 
31923 
31951 
31979 


.94924 
.94915 
.94906 
.94897 
.94888 

.94878 
.94869 
.94860 
.94851 
.94842_ 
.94832 
.94823 
.94814 
94805 
.94795_ 

.94786' 
.94777 

94768 
.94758 

94749 


33136 
33169 
33201 
33233 
33266 

33298 
33330 
33363 
33395 
33427_ 
. 33460 
33492 
33524 
33557 
33589 

33621 
33654 
33686 
33718 
33751 


3 
3 
3 
3 
3 


01783 
01489 
01196 
00903 
00611 


33106 
33134 
33161 
33189 
.33216_ 

33244 
33271 
33298 
33326 
33353 

33381 
33408 
33436 
33463 
33490 

33518 
33545 
33573 
33600 
33627 


94361 
94351 
94342 
94332 
.94322 

^94313 ~ 
.94303 
•94293 
•94284 
.94274_ 
94264 
94254 
94245 
94235 
94225_ 
94215 
94206 
94196 
•94186 
•94176 


.35085 
35118 
35150 
35183 

.35216 

.35248 
.35281 
.35314 
.35346 
.35379 
35412 
.35445 
•35477 
•35510 
.35543 
•35576 
•35608 
•35641 
35674 
35707 


2 
2 
2 
2 
2 

2 
2 

2 
2 
2 


•85023 
•84758 
•84494 
•84229 
83965 

•83702 
83439 
83176 
82914 

•82653 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


3 
3 

2 
2 
2 


00319 
00028 
99738 
99447 
99158 


35 
34 
33 
32 
31 


ao 

31 
32 
33 
34 
35 
36 
37 
38 
39 


2 

2 
2 

2 

2 
2 
2 
2 
2 
2 


98868 

98580 

98292 

98004 

97717_ 

97430 

97144 

96858 

96573 

96288 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


•82391 
•82130 

81870 
•81610 

81350 

81091 
80833 
80574 
80316 
80059 


30 

29 

28 
27 

ne 

25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


32006 
32034 
32061 
32089 
•32116 
.32144 
.32171 
32199 
.32227 
32254 
32282 
32309 
32337 
32364 
32392 
32419 
32447 
32474 
32502 
.32529__ 
32557 


94740 
94730 
94721 
•94712 
•94702 
•94693 
94684 
94674 
94665 
94656 
94646 
•94637 
•94627 
.94618 
.94609 

•94599 
•94590 
•94580 

94571 
.94561_ 

94552 


33783 
33816 
33848 
33881 

.33913_ 
33945 
33978 
34010 
34043 
34075_ 
34108 
34140 
34173 
34205 
34238 
34270 
34303 
34335 
34368 
34400 

•34433 


2 
2 
2 
2 
2 


96004 
95721 
95437 
95155 
94872 


33655 
33682 
33710 
•33737 
33764 


•94167 
•94157 
94147 
•94137 
.94127_ 
•94118 
94108 
•94098 
94088 
•94078 
•94068 
94058 
94049 
94039 
_94029_ 

94019 
94009 
•93999 
•93989 
•93979 
93969 


35740 
35772 
35805 
•35838 
35871 


2 
2 
2 
2 
2 


79802 
79545 
79289 
79033 
78778 


'40 

19 
18 
17 
16 


45 
46 
47 
48 
49 


2 
2 
2 
2 
2 


94591 
94309 
94028 
93748 
93468 


33792 
33819 
33846 

•33874 
33901 
33929 
33956 

•33983 

•34011 
34038 

•34065 
34093 
34120 

•34147 
34175 
34202' 


.35904 
•35937 
.35969 
•36002 
36035 

36068 

36101 
.36134 
•36167 

36199 
•36232 
•36265 
.36298 
•36331 

36364 


2 
2 
2 

2 
2 


78523 
78269 
78014 
77761 
77507 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


2 
2 
2 
2 
2 


93189 
92910 
92632 
92354 
92076 


2 
2 
2 
2 
2 


77254 
77002 
76750 
76498 
76247 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


2 
2 

2 
2 
2 


91799 
91523 
91246 
90971 
90696 


2 
2 
o. 

2. 
2. 


75996 
75746 
75496 
75246 
74997 


5 

4' 
3 
2 
1 


<U» 


2 


90421 


36397 


2. 


74748 





' 


Cos. 


Sin. 


C\)t. 


Tan. 1 


C(),<. Sin. 1 


Cot. 


Tan. I ' 



092 



.70° 



'rAHLM l\.- N.\ I liL'ALSrNrRS.CYXmNEH.TANOIONTH, ANI)<orAN(;i;N IS. 
'Hi' ii 



'" 


Sill. 


C.i. 


' 1"; 1 1 1 . 


Cnl,. 


Sill. 


( '(I . 


r.-iii. 


( '.,( 







34202 


i)3!)(n) 


•30397 


2 74 748 


35837 


!)33!)H 


30380 


:: 001)09 


<;o 


1 


.342:!{) 


l)31l!)l) 


30430 


2-74499 


35804 


• 9.334 


384 UO 


:: 00203 


!)!) 


2 


.34:!(>7 


1)31)41) 


30403 


2. 74251 


35891 


.93337 


384 53 


l^ 00057 


!)H 


3 


34284 


931)31) 


30490 


2 74004 


35918 


.9331i7 


38487 


2. 59831 


57 


J 


3431 J 


1)3929 


■ 'MWi'JM 


:i 73750 


3594 5 


93310 


.385iiO 


2 59000 


50 


B 


34:{;iii 


93919 


301)02 


'J,- 73!)09 


3!) 9 73 


!)3:{00 


30 553 


2 59381 


55 


(i 


;:]:•:(.(. 


1)31)09 


30595 


:^.73:;(»3 


;!()()()() 


!);'.:;:)!) 


30 '.a 7 


:: !)9150 


54 


7 


;m,;i);; 


93899 


30028 


:>.73()I7 


;i(i(j::7 


!);i:;hI) 


.3H(.:;o 


:M)8932 


53 


8 


■ M'VM 


93889 


30001 


i: 7::7yi 


3»)054 


93:; 74 


3001)4 


2 !)H708 


f)2 


9 


.34448 
.344yf) 


93879 
93899 


30094 


v. 7::.'.i:o 


30081 


93:i04 


30007 

307:; 1 


1 :; f)H4 84 

:: 58::oi 


51 


10 


307:^7 


- y:::;Hi 


30108 


!)3L'53 


r,(> 


11 


.34(j(J3 


938[>9 


.30700 


2- 7:;o30 


30135 


!)3:;4 3 


38754 


2. 58038 


49 


12 


.34(j30 


93849 


30793 


2 71792 


30102 


!)3:;3:; 


30787 


::.578i5 


48 


13 


34l)r)7 


93839 


30820 


li. 71548 


30190 


.932:;2 


30821 


2. 57593 


47 


14 


_34f)84 
34H12 


93829 
93819 


30859 
•30892 


li. 71305 
2. 71002 


302 r/_ 

.30244 


93211 
93201 


3HH!)4 
38888 


:: 57371 


40 


16 


2 57150 


41) 


16 


34 031) 


. 93809 


.30925 


li. 70819 


.3(Kr/i 


!i:{ M)0 


38921 


2 50928 


44 


17 


34(1()« 


.1)3799 


30958 


•i-myn 


.30::'))! 


!);: IHO 


38955 


:: 50707 


43 


18 


34(H)4 


93789 


30991 


li. 70335 


.3 03:; 5 


.93109 


38988 


:: 50487 


4 2 


11) 


34 7:M 


1)3 779 


.37o:m 


:> 70094 


303 5:: 


93 1 59 


39o:!:! 


:; i)0::oo 


41 


•^o 


34 74 8 


.937119 


37057 


1>.(;9H53 


30379 


93 14H 


39055 


;; 1)0040 


H) 


21 


34 77^) 


.9371)9 


37090 


:>.0!)(ii:; 


3(1-100 


.!)3I37 


39039 


:; 55h::7 


39 


22 


34 803 


9374 8 


37123 


i>.0!);i7l 


;i(.);i4 


93 i:; 7 


3!)!:::: 


:: 55008 


38 


23 


34 830 


.93 738 


37157 


:>.0!)I3I 


31)10 1 


93 1 I 


39 150 


:; !).53H9 


37 


24 


348r)7 
34884 


.93728 
93718 


.37 no 
.37:::;3 


:: (i8892 


30488 


. 93 1 00 
93095 


39190 


:; 5 5 170 


3() 


2S 


:j. 08053 


30515 


39223 


:; \A\)w:. 


3 5 


26 


341)12 


93708 


3 71} 50 


:! 084 14 


30542 


93084 


39:^57 


::. 54 734 


34 


27 


.34!)3I) 


.93098 


.37::89 


i;. 08 175 


.30509 


.93074 


• 39::90 


2.54 5lfi 


33 


28 


.34!) (Ml 


.93088 


.37322 


1^.07937 


.30590 


.93003 


• 393:;4 


::. 54 299 


32 


20 


.341)5)3 

.3f)():n 


.93077 


•37355 


2.07700 


.30o:;3 


.9305'3 


.39357 


::.54()h:: 


31 


:to 


.93007 


.37388 


2.074 02 


.30050 


.9304 2 


393!) 1 


:!.53H05 


M\ 


:;i 


.3f)04H 


.930 V/ 


.374:>2 


2.07:;::5 


.3i;077 


.!)3()3I 


.394:; 5 


:;. 5304 8 


21) 


,' '> " 


;{')()7.') 


93047 


37455 


1! O(.')H') 


;5H7()4 


'),'.():;() 


,'.') \u\\ 


:; 53432 


i:8 




:{')i():: 


93037 


37488 


1! ooy')'.; 


3(i7;:i 


!)3i)l() 


39'i'):; 


:; 53:: 17 


::7 




3f)l30 

3r)ir)7 


930 '.Mi 
.93010 


:r/ '.■.:! 


:: o(;5i(; 


3075H 


.!):M)99 


39'.:;o 


:; 5.3001 


::o 




.•{■/S'.'l 


1! Oli'.!Hl 


30785 


!) 1^988 


3!)')')!) 


:; 1)2780 


25 


.lit 


3'HH4 


930()(i 


3 7') OH 


.! (i004 


30Hi:! 


9:!!) 78 


39 ')•):; 


:; 5::57i 


24 


37 


:{'.:: 1 i 


!);{')!)i; 


37(i::i 


:: 0581 1 


.3f;H:{') 


9:!'H,7 


;{')(.;;(; 


:; f)::357 


23 


38 


;{'.:;:{!) 


!)3.')Jlf) 


37054 


V. 05!) 70 


3 OH (17 


•):;').'.(, 


,',')(, (.0 


:; 5:: 142 


22 


3J> 


■AW.>\A\ 


!)3')7f. 


3 7flH7 


'.'.■ 05342 


3(;H94 


9:!!)4f) 


3!M1!M 


:; 519::!) 


') 1 


lO 


3f):!!)3 


.93!)0f) 


37720 


:».(i5l09 


30921 


9:M)35 


397:r/ 


:: 51715 


'*A\ 


11 


3r)32() 


.93f)f)f) 


377f)4 


2 04 875 


3094 8 


9:!9:!4 


39701 


:; 5 150:: 


19 


lli 


3rj347 


.931)44 


.{7707 


:: 04 04 2 


30!) 7 5 


'):!')I3 


;;')7')'. 


:; '.i::)!9 


18 


lit 


3r)37r) 


.9 3.') 3 4 


37h::() 


:: 044 10 


370o:: 


'j:;i)():; 


:i!)H:;!) 


:; 5 107(1 


17 


l-l 


3.^)402 


.\yM)?A 


3 70. 53 


V. 04 1 77 


370::9 


9:! 09:: 

9:^88 1 


3!) ho:: 

39890 


:: !)OH04 
:!.5O05:: 


10 


-If) 


3r)4 29 


.93514 


37887 


2 0394 5 


37050 


15 


-1(1 


3r)4r)H 


.93503 


379:!0 


2 03714 


37083 


9'.>870 


3f)930 


:: 1)0440 


14 


17 


3r)4 84 


93493 


37953 


2 03483 


371 10 


92859 


3 99 '13 


: so:;:;9 


13 


48 


:\uu\ 1 


934 83 


37980 


2 03252 


37137 


92849 


39!)97 


:; ')()() 10 


12 


49 


3f)f)38 

3r)r)nr> 


.934 72 
.03402 


380:^0 
38053 


2 03021 


37104 


9'^838 


4003 1 


:: 49H07 


11 


M) 


2.02791 


37191 


\y.\mi 


40005 


2-49597 


lO 


ni 


3')f>02 


934 52 


38080 


2.0:>50l 


37218 


9:^810 


40098 


2 49380 


1) 




MMiM) 


9344 1 


30i:!O 


:>. fi :!:{;{■.! 


3 7:^4 5 


9:'H()5 


40132 


•: 49 177 


8 




:;m1'17 


934 3 1 


30153 


'.! o::i()3 


W'lW'IW 


').!794 


40100 


:: 4 09 1.7 


7 


IjI 


3'j(l74 
35701 


934 IM) 


30 1 HO 


:: OIH74 


3 7:;!)!) 


!):;7H4 


40200 


2 4H7f)H 


fl 


56 


.934 10 


MV.V.'M 


?. I 04 


37320 


92773 


40234 


2 4 854 9 


Ti 


Bfi 


35728 


.93400 


:ui'.;f)3 


2 014 18 


373 53 


91^702 


40207 


2 48340 


4 


57 


.35755 


93389 


Mv.\m 


2 01 190 


37380 


9:: 751 


40301 


2 48132 


3 


68 


35782 


93379 


38320 


2 00903 


37407 


9:!740 


40335 


:: 4 7924 


2 


59 


35810 
3r)H37 


93308 
93358 


3H353 
38380 


V. 00730 
?. 00 50!) 


37434 


w.vnw 


40309 


:: 4 7710 


Jl 


((O 


3 7401 


92718 


40403 


:: 4 7509 


V 


' 


( '.. . 


Sill. 


Cnl. 


1 nil. 


( 'm . 


Sill. 


Col. 


Inn. 





093 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTa 
33° 23^ 



' 


Sin. 


Cos. 


Tan. 


Cot. 1 


Sin. 


Cos. 


Tan. 


€ot. 


' 




1 

2 
3 
4 


•37461 
■37488 
.37515 
•37542 
•37569 


•92718 
.92707 
•92697 
.92686 
.92675 


.40403 
.40436 
.40470 
.40504 
.40538 


2 

2 
2 
2 
2 


47509 
47302 
47095 
46888 
46682 


39073 
39100 
.39127 
•39153 
39180 


■92050 
.92039 
.92028 
.92016 
.92005_ 

.91994 
.91982 
•91971 
•91959 
•91948 


•42447 
•42482 
•42516 
.42551 
42585 


2-35585 
2-35395 
2-35205 
2-35015 
2.34825 

2.34636 
2.34447 
2.34258 
2.34069 
2-33881 


60 

59 
58 

57 

5r> 


5 
6 
7 
8 
9 


•37595 
•37622 
•37649 
•37676 
•37703 


.92664 1 
•92653 
.92642 
.92631 
92620 


.40572 
.40606 
.40640 
.40674 
.40707 


2 
2 
2 
2 
2 


46476 
46270 
46065 
45860 
45655 


.39207 
•39234 
•39260 1 
•39287 
-39314 


•42619 
•42654 
•42688 
.42722 
•42757 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


•37730 
•37757 
•37784 
•37811 
37838 


•92609 
.92598 
.92587 
.92576 ! 
.92565 : 


.40741 
.40775 
.40809 
.40843 
.40877 


2 
2 
2 
2 
2 


45451 
45246 
45043 
44839 
44636 


.39341 
.39367 
.39394 
•39421 
.39448 


•91936 
•91925 
•91914 
.91902 
^9181^1_ 
.91879 
.91868 
.91856 
•91845 
91833 


.42791 
.42826 
.42860 
.42894 
.42929 


2-33693 
2^33505 
2.33317 
2.33130 
2.32943 
2-32756 
2^32570 
2.32383 
2.32197 
2-32012 


50 

49 
48 
47 
46 


15 
16 
17 
18 

19 


•37865 
•37892 
•37919 
•37946 
37973 


.92554 
•92543 
•92532 
.92521 
92510 ! 


.40911 
.40945 
.40979 
.41013 
.41047 

.41081 
.41115 
.41149 
.41183 
.41217 


2 
2 
2 
2 

2 


44433 
44230 
44027 
43825 
43623 


.39474 
39501 
39528 

.39555 
39581 


.42963 
.42998 
.43032 
.43067 
.43101 


"45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.37999 
.38026 
.38053 
.38080 
38107 


.92499 1 
.92488 
.92477 
.92466 
.9245_5_ 
.92444 
.92432 : 
.92421 
.92410 
.92399_ 
.92388 
.92377 ' 
92366 ' 
.92355 
.92343 


2 
2 
2 
2 
2 


43422 
43220 
43019 
42819 
42618 


39608 
39635 
39661 
.39688 
•39715 


•91822 
•91810 
•91799 
.91787 
.91775 
•91764 
•91752 
•91741 
•91729 
.91718 

91706 
91694 
•91683 
•91671 
.91660 


•43136 
.43170 
.43205 
.43239 
.43274 
.43308 
•43343 
•43378 
.43412 
.43447: 
.43481 
•43516 
•43550 
•43585 
•43620 


2.31826 
2.31641 
2.31456 
2.31271 
2.31086 
2.30902 
2.30718 
2.30534 
2.30351 
2-30167 
2.29984 
2.29801 
2-29619 
2-29437 
2.29254 

2.29073 
2.28891 
2-28710 
2-28528 
2-28348 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•38134 
.38161 
38188 
.38215 
.38241 


.41251 
.41285 
41319 
.41353 
.41387_ 
.41421 
.41455 
.41490 
.41524 
.41558 


2 
2 
2 
2 
2 


42418 
42218 
42019 
41819 
41620 


39741 
39768 
39795 
39822 
39848 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.38268 
.38295 
.38322 
.33349 
.38376 


2 
2 
2 
2 
2 


41421 
41223 
4a025 
40827 
40629 


39875 
39902 
•39928 
•39955 
•39982 
•40008 
•40035 
40062 
40088 1 
40115 1 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.38403 
.38430 
•38456 
.38483 
.38510 


.92332 
.92321 ; 
.92310 
.92299 
.92287 


.41592 
.41626 
.41660 
.41694 
.41728 


2 
2 
2 
2 

2 


40432 
40235 
40038 
39841 
39645 


.91648 
.91636 
•91625 
•91613 
.91601 


.43654 
.43689 
.43724 
.43758 
.43793 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•38537 .92276 1 
•38564 .92265 
•38591 .92254 
.38617 .92243 
.38644 .92231 


.41763 
.41797 
.41831 
.41865 
.41899 


2 
2 
2 
2 
2 


39449 
39253 
39058 
38863 
38668 


40141 \ 
•40168 

40195 
•40221 
•40248 i 


-91590 
.91578 
.91566 
.91555 
.91543 

.91531 
.91519 
•91508 
.91496 
91484 


.43828 
.43862 
•43897 
•43932 
•43966 


2^28167 
2^27987 
2^27806 
2^27626 
2^27447 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•38671 
.38698 
,3«725 
.38752 
38778 


.92220 
.92209 
.92198 
.92186 
.92175 


.41933 
.41968 
.42002 
.42036 
.42070 


2 
2 
2 
2 
2 


38473 
38279 
38084 
37891 
37697 


•40275 
40301 
•40328 
•40355 
•40381 


•44001 
•44036 
•44071 
.44105 
-44140 
.44175 
-44210 
.44244 
.44279 
.44314 
.44349 
•44384 
.44418 
.44453 
. 44488 


2^27267 
2-27088 
2-26909 
2^26730 
2-26552 
2-26374 
2-26196 
2^26018 
2-25840 
2 •25663 
2.25486 
2-25309 
2-25132 
2-24956 
2-24780 
2^24604 
Tan. 


15 

14 
13 
12 
11 


50 

51 
52 
53 
54 


38805 
38832 
38859 
.38886 
.38912 


.92164 
.92152 
.92141 
.92130 
.92119 

.92107 
.92096 
.92085 
.92073 
.92062 


.42105 
.42139 
.42173 
.42207 
.42242 


2 
2 
2 
2 
2 


37504 
37311 
37118 
36925 
36733 


•40408 ' 
•40434 
40461 
•40488 
•40514 


.91472 
.91461 
91449 
•91437 
•91425 
•91414 
•91402 
•91390 
•91378 ' 
.91366 i 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


38939 
•38966 
•38993 
•39020 
•39046 


.42276 
.42310 
.42345 
.42379 
.42413 


2 
2 
2 
2 
2 


36541 
36349 
36158 
35967 
35776 


.40541 
•40567 
•40594 
•40621 
•40347 


5 
4 
3 
2 
1 


60 


•39073 


.92050 


.42447 


2 


35585 


•40674 


.91355 


•44523 





' 


Cos. 1 Sin. 1 


Cot. 


Tan. 1 


Cos. 


Sin. 1 


Cot. 


/ 



67° 



694 



66° 



TABLE IX.— NATURAL SINES. COSINES, TANGENTS. AND COTANGENTS. 
34° 25° 



/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 
4 


•40674 
.40700 
.40727 
.40753 
.40780 


•91355 
.91343 
.91331 
.91319 
.91307 


•44523 
•44558 
•44593 
•44627 
•44662 


2.24604 
2.24428 
2.24252 
2.24077 
2.23902 


.42262 
.42288 
.42315 
.42341 
.42367 


•90631 
•90618 
•90606 
•90594 
•90582 


.46631 
.46666 
.46702 
•46737 
•46772 


2.14451 
2.14288 
2.14125 
2.13963 
2.13801 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.40806 
.40833 
.40860 
.40886 
.40913 


.91295 
.91283 
.91272 
•91260 
.91248 


•44697 
•44732 
•44767 
•44802 
.44837 


2.23727 
2.23553 
2.23378 
2.23204 
2.23030 


.42394 
.42420 
.42446 
.42473 
.42499 


•90569 
.90557 
.90545 
.90532 
.90520 


.46808 
.46843 
.46879 
.46914 
.46950 


2.13639 
2.13477 
2.13316 
2.13154 
2.12993 


55 
54 
53 
52 
51 


10 

11 
12 
13 

14 


.40939 
.40966 
.40992 
•41019 
.41045 


•91236 
.91224 
•91212 
•91200 
•91188 


.44872 
.44907 
•44942 
•44977 
•45012 


2.22857 
2.22683 
2.22510 
2.22337 
2.22164 


.42525 
•42552 
.42578 
.42604 
.42631 


.90507 
.90495 
•90483 
•90470 
.90458 


.46985 
.47021 
.47056 
.47092 
.47128 


2.12832 
2.12671 
2.12511 
2.12350 
2. 12190 


50 

49 
48 

47 
46 


15 
16 
17 
18 
19 


•41072 
•41098 
.41125 
•41151 
.41178 


•91176 
•91164 
.91152 
•91140 
.91128 


.45047 
.45082 
•45117 
•45152 
•45187 


2.21992 
2.21819 
2.21647 
2.21475 
2.21304 


.42657 
.42683 
.42709 
.42736 
•42762 

.42788 
.42815 
•42841 
•42867 
.42894 


•90446 
•90433 
•90421 
. 90408 
•90396 

•90383 
•90371 
.90358 
.90346 
•90334 


.47163 
.47199 
.47234 
.47270 
.47305 

.47341 
.47377 
.47412 
.47448 
•47483 


2.12030 
2.11871 
2.11711 
2-11552 
2.11392, 

2.11233 
2.11075 
2.10916 
2.10758 
2. 10600 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


•41204 
•41231 
•41257 
.41284 
•41310 


.91116 
•91104 
•91092 
.91080 
91068 


•45222 
•45257 
•45292 
.45327 
•45362 


2.21132 
2.20961 
2.20790 
2.20619 
2.20449 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•41337 
•41363 
•41390 
•41416 
•41443 


.91056 
.91044 
.91032 
•91020 
.91008 


•45397 
•45432 
•45467 
•45502 
•45538 


2.20278 
2.20108 
2.19938 
2.19769 
2.19599 


.42920 
•42946 
•42972 
•42999 
.43025 


.90321 
.90309 
.90296 
•90284 
.90271 


•47519 
•47555 
.47590 
.47626 
•47662 


2.10442 
2.10284 
2.10126 
2.09969 
2.09811 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.41469 
.41496 
.41522 
.41549 
.41575 


.90996 
.90984 
.90972 
.90960 
•90948 
•90936 
.90924 
.90911 
90899 
.90887 


•45573 
•45608 
.45643 
.45678 
.45713 


2.19430 
2.19261 
2.19092 
2. 18923 
2.18755 


.43051 
.43077 
.43104 
.43130 
.43156 


.90259 
• 90246 
•90233 
•90221 
.90208 


.47698 
.47733 
.47769 
•47805 
.47840 


2.09654 
2.09498 
2-09341 
2.09184 
2^09028 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.41602 
•41628 
•41655 
.41681 
•41707 


.45748 
.45784 
.45819 
•45854 
•45889 


2.18587 
2.18419 
2.18251 
2.18084 
2.17916 


.43182 
.43209 
.43235 
.43261 
.43287 


.90196 
.90183 
.90171 
.90158 
.90146 


.47876 
.47912 
•47948 
•47984 
.48019 


2^08872 
2^08716 
2.08560 
2.08405 
2 08250 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•41734 
.41760 
•41787 
.41813 
41840 


.90875 
.90863 
•90851 
•90839 
90826 


•45924 
.45960 
.45995 
•46030 
•46065 


2 17749 
2.17582 
2.17416 
2.17249 
2.17083 


.43313 
.43340 
.43366 
•43392 
•43418 


.90133 
.90120 
.90108 
.90095 
.90082 


.48055 
.4809i 
•48127 
.48163 
.48198 


2.08094 
2.07939 
2-07785 
2-07630 
2.07476 

2.07321 
2.07167 
2.07014 
2 06860 
2- 06706 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.41866 
•41892 
•41919 
.41945 
•41972 


.90814 
•90802 
.90790 
•90778 
•90766 


.46101 
.46136 
.46171 
•46206 
•46242 


2.16917 
2.16751 
2.16585 
2.16420 
2. 16255 


. 43445 
•43471 
.43497 
•43523 
•43549 


.90070 
.90057 
.90045 
.90032 
.90019 


.48234 
.48270 
•48306 
.48342 
•48378 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


41998 
.42024 

42051 
•42077 
.42104 


•90753 
•90741 
.90729 
.90717 
90704 


•46277 
•46312 
.46348 
.46383 
.46418 


2.16090 
2.15925 
2.15760 
2.15596 
2.15432 


•43575 
•43602 
•43628 
.43654 
43680 


.90007 
.89994 
.89981 
89968 
•89956 


.48414 
.48450 
.48486 
.48521 
48557 


2^06553 
2.06400 
2.06247 
2.06094 
2.05942 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•42130 
•42156 
.42183 
•42209 
•42235 


.90692 
•90680 
•90668 
.90655 
.90643 


.46454 1 2.15268 
.46489 1 2.15104 
.46525 \ 2.14940 
.46560 2 14777 
•46595 1 214614 


43706 
•43733 
43759 
43785 
43811 


•89943 
89930 
•89918 
•89905 
•89892 


•48593 
•48629 
.48665 
.48701 
.48737 


2^05790 
2 •05637 
2.05485 
2.05333 
2.05182 


5 

4 
3 
2 
1 


22- 


•42262 


.90631 


•46631 2.14451 


43837 


.89879 


.48773 


2.05030 


_2 


— /— 


Cos. 


Sin. 


Cot. Tan. 


Cos. 

1 


Sin. 


Cot. 


Tan. 


/ 



65° 



695 



64° 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
26° 27° 



t 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 
4 


•43837 
•43863 
•43889 
•43916 
.439^2 


.89879 
.89867 
.89854 
.89841 
.89828 


.48773 

.48809 

.48845 

.48881 

.48917_ 

.48953 

.48989 

.49026 

.49062 

.49098 


2.05030 
2-04879 
2.04728 
2.04577 
2^04426 
2^04276 
2.04125 
2.03975 
2.03825 
2.03675 


.45399 
-45425 
-45451 
-45477 
-45503 


.89101 
.89087 
.89074 
.89061 
-89048 


-50953 
-50989 
-51026 
.51063 
.51099 




96261 
.96120 
.95979 
.95838 
.95698 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


•43968 
•43994 
•44020 
• 44046 
•44072 
•44098 
•44124 
•44151 
•44177 
•44203 


.89816 
.89803 
.89790 
.89777 
.89764 


-45529 
-45554 
.45580 
•45606 
.45632 


-89035 
.89021 
.89008 
.88995 
.88981 


.51136 
.51173 
.51209 
.51246 
.51283_ 
.51319 
-51356 
-51393 
•51430 
•51467 




-95557 
-95417 
-95277 
-95137 
•94997 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.89752 
.89739 
.89726 
•89713 
.89700 

.89687 
.89674 
.89662 
•89649 
.89636 


•49134 
.49170 
•49206 
.49242 
.49278 
.49315 
.49351 
•49387 
•49423 
.49459 


2.03526 
2.03376 
2.03227 
2.03078 
2-02929 


•45658 
•45684 
•45710 
•45736 
.45762 


.88968 
-88955 
-88942 
.88928 
.88915 
.88902 
-88888 
.88875 
.88862 
.88848 




-94858 
-94718 
-94579 
. 94440 
.94301 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


•44229 
•44255 
•44281 
•44307 
•44333 


2.02780 
2^02631 
2 •02483 
2^02335 
202187 


.45787 
.45813 
.45839 
-45865 
.45891 


-51503 
-51540 
.51577 
.51614 
•51651 




.94162 
.94023 
.93885 
•93746 
.93608 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


.44359 
.44385 
•44411 
•44437 
. 44464 


•89623 

•89610 

•89597 

.89584 

j_89571^ 

•89558 

•89545 

•89532 

•89519 

.89506 

•89493 
•89480 
•89467 
.89454 
.89441 
.89428 
.89415 
.89402 
.89389 
•89376 


.49495 
.49532 
.49568 
•49604 
.49640 


2^02039 
2^01891 
2^01743 
2.01596 
2 01449 


-45917 
•45942 
-45968 
45994 
-46020 


.88835 
.88822 
.88808 
•88795 
88782 


•51688 
.51724 
•51761 
.51798 
•51835 




.93470 
.93332 
.93195 
-93057 
.9?920 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•44490 
•44516 
•44542 
.44568 
.44594^ 

.44620 
•44646 
•44672 
44698 
44724 
.44750 
.44776 
.44802 
44828 
44854 


.49677 
.49713 
•49749 
.49786 
•49822 


2.01302 
2.01155 
2-01008 
2-00862 
2. 00715 


-46046 
-46072 
-46097 
.46123 
46149 


•88768 
.88755 
.88741 
.88728 
.88715 


•51872 
.51909 
•51946 
•51983 
.52020 




-92782 
•92645 
.92508 
.92371 
.92235 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


•49858 
•49894 
.49931 
•49967 
.50004 


2-00569 
2 00423 
2-00277 
2-00131 
1-99986 


.46175 
.46201 
•46226 
•46252 
•46278 


.88701 
.88688 
.88674 
•88661 
•88647 


•52057 
•52094 
•52131 
-52168 
-52205 




.92098 
-91962 
.91826 
•91690 
91554 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


•50040 
•50076 
.50113 
.50149 
.50185 


1.99841 
1.99695 
1-99550 
1-99406 
1-99261 


•46304 
•46330 
-46355 
•46381 
•46407 


•88634 
.88620 
.88607 
.88593 
.88580 


.52242 
.52279 
.52316 
.52353 
.52390 




.91418 
.91282 
.91147 
.91012 
90876 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•44880 

.44906 

.44932 

.44958 ! 

.44984 

• 45010 

•45036 

.45062 

•45088 

•45114 


.89363 
.89350 
•89337 
.89324 
•89311 


•50222 
.50258 
.50295 
•50331 
•50368 


1-99116 
1-98972 
1-98828 
1-98684 
1-98540 


•46433 
•46458 
•46484 
•46510 
•46536 


•88566 
•88553 
•88539 
•88526 
•88512 


-52427 
-52464 
-52501 
-52538 
-52575 




90741 
90607 
90472 
90337 
90203 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•89298 
.89285 
•89272 
•89259 
.89245 


• 50404 
•50441 
•50477 
•50514 
.50550 


1-98396 
1-98253 
1-98110 
1-97966 
1-97823 


•46561 
•46587 
.46613 
•46639 
•46664 


•88499 
.88485 
•88472 
•88458 
.88445 


-52613 
•52650 
.52687 
•52724 
.52761 




90069 
89935 
89801 
89667 
89533 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•45140 
•45166 
•45192 
.45218 
.45243 


.89232 
.89219 
.89206 
•89193 
.89180 


.50587 
.50623 

50660 
•50696 
•50733 
• 50769 
•50806 
•50843 
.50879 

50916 


1.97681 
1.97538 
1.97395 
1.97253 
1.97111 


.46690 
•46716 
.46742 
.46767 
.46793 


.88431 
.88417 
•88404 
•88390 
.88377 


•52798 
•52836 
•52873 
.52910 
•52947 




89400 
89266 
89133 
89000 
88867 


10 

9 
8 

7 
6 


55 
56 
57 
58 
59 


•45269 
•45295 
•45321 
•45347 
•45373 


89167 
•89153 
.89140 

89127 
.89114 


1-96969 
1-96827 
1. 96685 
1^96544 
1-96402 


.46819 
.46844 
•46870 
-46896 
-46921 


•88363 
-88349 
.88336 
.88322 
.88308 


•52985 
•53022 
•53059 
-53096 
•53134 




88734 
88602 
88469 
88337 
88205 


1 

2 

1 


60 


•45399 


.89101 


•50953 


1.96261 


-46947 


-88295 


-53171 


1_- 


88073 





' 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. Cot. 


Tan. 





63° 



696 



62° 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
28° 39° 



f 


Sin. 


Cos. 


Tan. 


Cot. 1 


Sin. 


Cos. 


Tan. 


Cot. 


f 




1 

2 
3 
4 


.46947 
.46973 
.46999 
.47024 
.47050 


•88295 
•88281 
•88267 
•88254 
88240 


•53171 
.53208 

53246 
.53283 
.53320 
.53358 
.53395 
.53432 

53470 
•53507 




88073 
87941 
87809 
87677 
87546 


48481 
•48506 
•48532 
.48557 
•48583 


87462 
.87448 
.87434 

3^7420 
.87406 


•55431 
•55469 
.55507 
.55545 
.55583 


1 80405 
1.80281 
1.80158 
1.80034 
1.79911 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.47076 
•47101 
.47127 
.47153 
.47178 


88226 
88213 
•88199 
•88185 
88172 

•88158 
•88144 
•88130 
•88117 
•88103 




87415 
87283 
87152 
87021 
86891 


.48608 
.48634 
.48659 
48684 
•48710 


•87391 
•87377 
.87363 
.87349 
.87335 


•55621 
•55659 
•55697 
•55736 
•55774 


1 79788 
1 79665 
1.79542 
1.79419 
1.79296 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.47204 
.47229 
.47255 
.47281 
.47306 


•53545 
•53582 
.53620 
.53657 
•53694 




86760 
86630 
86499 
86369 
86239 


•48735 
•48761 
.48786 
.48811 
•48837 


.87321 
.87306 
.87292 
.87278 
.87264 


•55812 
•55850 
•55888 
.55926 
•55964 


1.79174 
1.79051 
1.78929 
1.78807 
1.78685 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19_. 


.47332 
•47358 
•47383 
•47409 
.47434 


.88089 
•88075 
.88062 
•88048 
.88034 


•53732 
.53769 
.53807 
.53844 
.53882 




86109 
85979 
85850 
85720 
85591 
85462 
85333 
85204 
85075 
84946 


.48862 
.48888 
.48913 
.48938 
•48964 

48989 
•49014 
•49040 
•49065 
•49090 


.87250 
.87235 
.87221 
.87207 
•87193 

.87178 
.87164 
•87150 
•87136 
.87121 


.56003 
.56041 
•56079 
.56117 
•56156 
•56194 
•56232 
•56270 
•56309 
•56347 


1.78563 
1.78441 
1-78319 
1-78198 
1^78077 
1^77955 
1^77834 
1-77713 
1^77592 
1^77471 


45 
44 
43 
42 


30 

21 
22 
23 
24 


•47460 
•47486 
•47511 
•47537 
.47562 


.88020 
88006 
•87993 
.87979 
•87965 


.53920 
.53957 
.53995 
.54032 
.54070 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


47588 
•47614 

47639 
.47665 
.47690 


•87951 
•87937 
•87923 
•87909 
.87896 


.54107 
.54145 
54183 
.54220 
.54258 




84818 
84689 
84561 
84433 
84305 


.49116 
•49141 
•49166 
•49192 
.49217 


•87107 
87093 
•87079 
•87064 
•87050 


•56385 
•56424 
•56462 
•56501 
•56539 


1^77351 
1. 77230 
1.77110 
1^76990 
1^ 76869 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.47716 
.47741 
.47767 
•47793 
•47818 


•87882 
•87868 
•87854 
•87840 
•87826 


.54296 
•54333 
•54371 

• 54409 

• 54446 




84177 
84049 
83922 
83794 
83667 


.49242 
•49268 
•49293 
.49318 
.49344 


•87036 
.87021 
.87007 
.86993 
•86978 


•56577 
•56616 
.56654 
.56693 
•56731 


1^76749 
1^76629 
1^76510 
r 76390 
1.76271 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.47844 
•47869 
.47895 
.47920 
.47946 


•87812 
.87798 
•87784 
•87770 
•87756 


• 54484 
•54522 
•54560 
•54597 
.54635 




83540 
83413 
83286 
83159 
83033 


.49369 
.49394 
•49419 
•49445 
•49470 


•86964 
•86949 
•86935 
•86921 
•86906 


•56769 
•56808 
•56846 
•56885 
•56923 


1.76151 
1.76032 
1.75913 
1^75794 
1^75675 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.47971 
.47997 
.48022 
.48048 
.48073 


•87743 
•87729 
.87715 
•87701 
•87687 


•54673 
•54711 
• 54748 
•54786 
.54824 




82906 
82780 
82654 
82528 
82402 


•49495 
•49521 
•49546 
•49571 
.49596 


•86892 
.86878 
.86863 
•86849 
•86834 


•56962 
•57000 
•57039 
•57078 
.57116 


1^75556 
1.75437 
1^75319 
1^75200 
1^75082 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•48099 
•48124 
•48150 
•48175 
•48201 


•87673 
•87659 
87645 
•87631 
.87617 


.54862 
.54900 
•54938 
.54975 
.55013 




82276 
82150 
82025 
81899 
81774 


•49622 
•49647 
•49672 
•49697 
•49723 


.86820 
•86805 
•86791 
•86777 
•86762 


•57155 
•57193 
•57232 
•57271 
•57309 


1^74964 
1 . 74846 
1^ 74728 
1^74610 
1^74492 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•48226 
•48252 
48277 
.48303 
.48328 


.87603 
.87589 
•87575 
•87561 
.87546 


.55051 
.55089 
.55127 
•55165 
.55203 




81649 
81524 
81399 
81274 
81150 


•49748 
•49773 
•49798 
•49824 
•49849 


•86748 
•86733 
•86719 
•86704 
•86690 


•57348 
•57386 
•57425 
■57464 
•57503 


1^74375 
1^74257 
1 • 74140 
1 • 74022 
1^73905 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.48354 
•48379 
.48405 
.48430 
.48456 


•87532 
•87518 
•87504 
•87490 
87476 


•55241 
.55279 
.55317 
.55355 
•55393 




81025 
80901 
80777 
80653 
80529 


•49874 
•49899 
•49924 
49950 
•49975 


•86675 
•86661 
•86646 
•86632 
•86617 


•57541 
•57580 
•57619 
•57657 
•57696 


1.73788 
1.73671 
1.73555 
1.73438 
1.73321 


5 
4 
3 
2 

1 


60 


.48481 

Cos. 


87462_ 
Sin. 


•55431 
Cot. 


.1. 


80405 


• 50000 


.86603 


•57735 


1.73205 









Tan. 


Coso 


Sin. 


Cot. 


Tan. 


' 



61° 



697 



60° 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
30° 31° 



' 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 1 


' 




1 

2 
3 

4 


.50000 
.50025 
.50050 
.50076 
.50101 


•86603 
•86588 
•86573 
.86559 
•86544 


57735 
57774 
57813 
•57851 
.57890 




73205 
73089 
72973 
72857 
72741 


•51504 
•51529 
.51554 
.51579 
•51604 


85717 
85702 
•85687 
•85672 
•85657 


.60086 
.60126 
•60165 
■60205 
•60245 




66428 
66318 
66209 
66099 
65990 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


•50126 
.50151 
.50176 
.50201 
.50227 


•86530 
•86515 
.86501 
•86486 
.86471 


.57929 
•57968 
.58007 
.58046 
•58085 




72625 
72509 
72393 
72278 
72163 


•51628 
•51653 
•51678 
•51703 
•51728 

•51753 
•51778 
•51803 
•51828 
.51852 


•85642 
•85627 
•85612 
•85597 
•85582 

■85567 
•85551 
•85536 
•85521 
•85506 


•60284 
■60324 
•60364 
.60403 
.60443 




65881 
65772 
65663 
65554 
65445 


55 
54 
53 
52 
51 


10 

11 
12 
13 

14 


.50252 
•50277 
50302 
.50327 
.50352 


•86457 j 
•86442 
•86427 
•86413 
•86398 ' 


•58124 
•58162 
•5820] 
•58240 
58279 




72047 
71932 
71817 
71702 
71588 


.60483 
.60522 
.60562 
.60602 
■60642 




65337 
65228 
65120 
65011 
64903 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.50377 
. 50403 
50428 
50453 
50478 


•86384 ! 
•86369 
•86354 
•86340 
86325 


•58318 
•58357 
•58396 
•58435 
•58474 




71473 
71358 
71244 
71129 
71015 


•51877 
•51902 
•51927 
.51952 
.51977 


•85491 -60681 
•85476 -60721 
•85461 .60761 
•85446 -60801 
•85431 60841 




.64795 
•64687 
•64579 
•64471 
•64363 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


50503 
50528 ! 

•50553 ; 
50578 

•50603 


.86310 
•86295 
•86281 
•86266 
.86251 1 


•58513 
58552 
58591 
58631 

•58670 


-^ 


70901 
•70787 
•70673 
•70560 
• 7044:6_ 
70332 
70219 
70106 
69992 
69879 


52002 

52026 

•52051 

52076 

52101 1 

•52126 

•52151 

•52175 

•52200 

•52225 


•85416 -60881 
•85401 .60921 
•85385 60960 
•85370 61000 
.85355 61040 




•64256 
.64148 
•64041 
•63934 
.63826 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


50628 
•50654 
•50679 
50704 
50729 ' 


•86237 
.86222 
.86207 
•86192 
86178 


58709 
.58748 
•58787 
.58826 
•58865 


•85340 61080 

•85325 •61120 

•85310 61160 

85294 • 61200 

85279 ,• 61240 




•63719 
.63612 
.63505 
.63398 
•63292 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


•50754 
50779 
50804 

•50829 
50854 


.86163 
86148 

•86133 
86119 

•86104 


.58905 
.58944 
.58983 
.59022 
.59061 
•59101 
.59140 
•59179 
59218 
•59258 




69766 
69653 
69541 
69428 
69316 


•52250 
•52275 
52299 
•52324 
■52349 


85264 
85249 
85234 
•85218 
•85203 


•61280 
•61320 
.61360 
.61400 
•61440 




.63185 
•63079 
•62972 
•62866 
•62760 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


50879 

• 50904 

50929 

50954 

•50979 


•86089 
86074 
•86059 
•86045 
860.30 




69203 
69091 
68979 
68866 
68754 


■52374 
■52399 
•52423 
52448 
.52473 


•85188 
85173 
85157 
85142 
85127 


•61480 
•61520 
•61561 
•61601 
.61641 




•62654 
•62548 
•62442 
•62336 
•62230 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•51004 
•51029 
•51054 
•51079 
•51104 


.86015 
86000 
85985 
•85970 
•85956 


•59297 
•59336 
•59376 
.59415 
•59454 




68643 
68531 
68419 
68308 
68196 


52498 
•52522 
•52547 
•52572 
•52597 ■ 


•85112 
•85096 
•85081 
85066 
•85051 


•61681 
•61721 
•61761 
.61801 
■61842 

.61882 
•61922 
61962 
•62003 
•62043 




•62125 
.62019 
.61914 
•61808 
•61703 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•51129 
51154 
•51179 
• 51204 
51229 


•85941 i 
•85926 1 
•85911 
•85896 
•85881 


•59494 
•59533 
59573 
•59612 
•59651 




68085 
67974 
67863 
67752 
67641 


52621 ' 
.52646 
•52671 
•52696 
.52720 


85035 
•85020 
•85005 
84989 
84974 




-61598 
-61493 
•61388 
•61283 
-61179 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


• 51254 
•51279 
51304 
51329 
51354 


•85866 
•85851 
85836 : 
•85821 
•85806 


•59691 
59730 
•59770 
•59809 
•59849 




67530 
67419 
67309 
67198 
67088 


•52745 
•52770 
•52794 
•52319 
.52844 

•52869 
•52893 
•52918 
•52943 
•52967 


■84959 
. 84943 
■84928 
■84913 
84897 


•62083 
62124 
.62164 
.62204 
•62245 




-61074 
-60970 
60865 
-60761 
.60657 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•51379 
•51404 
.51429 
•51454 
.51479 


•85792 
•85777 
85762 : 
85747 
85732 j 


59888 
59928 
59967 
60007 
60046 




66978 
66867 
66757 
66647 
66538 


84882 
•84868 
•84851 
•84836 
•84820 


.62285 
•62325 
.62366 
•62406 
.62446 


I 


60553 
60449 
60345 
60241 
60137 


5 

4 
3 

2 

1 


go 


•51504 


■85717 1 


60086 


1 


66428 


.52992 1 
Cos. j 


84805 

Sin. 


.62487 
Cot. 


60033 





/ 


Cos. 1 Sin. 1 Cot. 


Tan. 


..Tan. 


' 



59° 



698 



58° 



TAI 


JLE IX.- 


-NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS 
32° 33° 


/ 


Sin. 


Cos 


Tan. 


Cot 1 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 

3 

,4 


.52992 
.53017 
.53041 
•53066 
.53091 


84805 
84789 
■84774 
84759 
84743 


62487 
.62527 
.62568 
.62608 
•62649 




60033 
59930 
59826 
59723 
59620 


. 54464 
. 54488 
•54513 
•54537 
•54561 


•83867 
•83851 
.83835 
.83819 
.83804 


•64941 
.64982 
.65024 
•65065 
.65106 




53986 
53888 
53791 
53693 
53595 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.53115 
.53140 
.53164 
•53189 
•53214 


84728 
•84712 
•84697 
•84681 
•84666 


•62689 
•62730 
.62770 
.62811 
.62852 




59517 
59414 
59311 
59208 
59105 


•54586 
•54610 
•54635 
54659 
54683 


.83788 
.83772 
.83756 
•83740 
.83724 


.65148 
.65189 
.65231 
.65272 
.65314 




53497 
53400 
53302 
53205 
53107 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


•53238 
•53263 
•53288 
.53312 
.53337 


•84650 
.84635 
.84619 
.84604 
.84588 


.62892 
.62933 
.62973 
•63014 
•63055 




59002 
58900 
58797 
58695 
58593 


54708 
•54732 
.54756 
.54781 
. 54805 


•83708 
•83692 
•83676 
.83660 
.83645 
•83629 
•83613 
.83597 
.83581 
.83565 


.65355 
•65397 
.65438 
.65480 
•65521 




53010 
52913 
52816 
52719 
52622 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


•53361 
.53386 
•53411 
.53435 
•53460 


.84573 
•84557 
•84542 
.84526 
•84511 


•63095 
•63136 
•63177 
.63217 
.63258 




58490 
58388 
58286 
58184 
58083 


•54829 
.54854 
.54878 
•54902 
•54927 


•65563 
•65604 
•65646 
•65688 
•65729 




52525 
52429 
52332 
52235 
52139 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.53484 
.53509 
.53534 
•52558 
.53583 


.84495 
.84480 
.84464 
.84448 
.84433 


.63299 
.63340 
•63380 
.63421 
.63462 




57981 
57879 
57778 
57676 
57575 


54951 
•54975 
.54999 
.55024 
.55048 


.83549 
.83533 
•83517 
.83501 
.83485 


•65771 
•65813 
•65854 
•65896 
•65938 




52043 
51946 
51850 
51754 
51658 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.53607 
•53632 
•53656 
53681 
.53705 


•84417 
.84402 
.84386 
•84370 
•84355 


.63503 

.63544 

63584 

63625 

.63666 




57474 
57372 
57271 
57170 
57069 


•55072 
.55097 
.55121 
.55145 
.55169 


•83469 
•83453 
.83437 
.83421 
.83405 


•65980 
•66021 
•66063 
•66105 
•66147 




51562 
51466 
51370 
51275 
51179 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


•53730 
•53754 
.53779 
.53804 
.53828 


•84339 
•84324 
•84308 
.84292 
•84277 
•84261 
.84245 
•84230 
.84214 
.84198 


•63707 
•63748 
•63789 
•63830 
•63871 
.63912 
•63953 
•63994 
•64035 
•64076 




56969 
56868 
56767 
56667 
56566 


.55194 
.55218 
•55242 
.55266 
.55291 


.83389 
.83373 
.83356 
.83340 
.83324 


•66189 
•66230 
•66272 
•66314 
.66356 




51084 
50988 
50893 
50797 
50702 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


•53853 
•53877 
•53902 
.53926 
.53951 




56466 
56366 
56265 
56165 
56065 


.55315 
.55339 
.55363 
•55388 
•55412 


.83308 
•83292 
•83276 
•83260 
.83244 


•66398 
.66440 
•66482 
•66524 
.66566 


1 


50607 
50512 
50417 
50322 
50228 

50133 
50038 
49944 
49849 
49755 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.53975 

• 54000 
•54024 

• 54049 
.54073 


•84182 
.84167 
.84151 
.84135 
.84120 

.84104 
•84088 
•84072 
.84057 
•84041 


.64117 
.64158 
.64199 
. 64240 
.64281 
.64322 
.64363 
. 64404 
. 64446 
.64487 




55966 
55866 
55766 
55666 
55567 


•55436 
.55460 
•55484 
•55509 
.55533 

•55557 
•55581 
•55605 
•55630 
55654 


.83228 
.83212 
•33195 
•83179 
•83163 


•66608 
•66650 
•66692 
•66734 
.66776 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.54097 
.54122 
.54146 
.54171 
.54195 




55467 
55368 
55269 
55170 
55071 


.83147 
.83131 
•83115 
.83098 
.83082 


•66818 
•66860 
•66902 
•66944 
.66986 




49661 
49566 
49472 
49378 
49284 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.54220 
.54244 
.54269 
.54293 
.54317 


•84025 
.84009 
83994 
83978 
83962 


.64528 
.64569 
.64610 
.64652 
.64693 


1 


54972 
54873 
54774 
54675 
54576 


.55678 
.55702 
•55726 
•55750 
.55775 


•83066 
■83050 
83034 
83017 
83001 


•67028 
•67071 
.67113 
•67155 
•67197 




49190 
49097 
49003 
48909 
48816 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.54342 
-54366 
.54391 
.54415 
• 54440 
. 54464 


.83946 
•83930 
.83915 
83899 
•83883 


•64734 
.64775 
.64817 
•64858 
64899 




54478 
54379 
54281 
54183 
54085 


•55799 
.55823 
•55847 
•55871 
•55895 


82985 
82969 
82953 
82936 
■82920 


•67239 
•67282 
•67324 
•67366 
.67409 




48722 
48629 
48536 
48442 
48349 


5 
4 
3 
2 
1 


60 


83867 


•64941 


1 


53986 


55919 


■82904 


.67451 


1 


48256 





' 


Cos. 


Sin. 


Cot. 


Tan. 1 


Cos. 


Sin. 


Cot. 


Tan. 1 


' 



57° 



699 



66** 



TAB 


JLE IX.- 


-NATURAL SINES, COSINES. TANGENTS, AND COTANGENTS 
34° 35° 




Sin. 


Cos. 


Tan. 


Cot. 1 


Sin. 


Cos. 


Tan. 


Cot. 1 


' 




1 
2 
3 
4 


.55919 
•55943 
•55968 
•55992 
.56016 


82904 
•82887 
•82871 
•82855 

82839 


•67451 
•67493 
•67536 
•67578 
-67620 




48256 
48163 
48070 
47977 
47885 


•57358 
•57381 
•57405 
•57429 
•57453 


.81915 
.81899 
•81882 
•81865 
•81848 


•70021 
•70064 
•70107 
•70151 
•70194 




42815 
42726 
42638 
42550 
42462 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 


5 
6 
7 
6 
9 


•56040 
•56064 
•56088 
.56112 
•56136 


•82822 
•82806 
•82790 
.82773 
•82757 


•67663 
•67705 
•67748 
•67790 
•67832 


1 


47792 
47699 
47607 
47514 
47422 


•57477 
•57501 
•57524 
•57548 
•57572 


•81832 
•81815 
•81798 
•81782 
•81765 


•70238 
.70281 
.70325 
•70368 
.70412 




42374 
42286 
42198 
42110 
42022 


10 

11 
12 
13 
14 


•56160 
.56184 
.56208 
•56232 
•56256 


•82741 
•82724 
•82708 
•82692 
.82675 


•67875 
•67917 
•67960 
•68002 
•68045 




47330 
47238 
47146 
47053 
46962 


.57596 
.57619 
•57643 
.57667 
•57691 


•81748 
.81731 
•81714 
81698 
81681 


.70455 
•70499 
.70542 
.70586 
.70629 




41934 
41847 
41759 
41672 
41584 


50 

49 
48 
47 
46 


15 

16 
17 
18 
19 


.56280 
.56305 
.56329 
•56353 
•56377 


.82659 
.82643 
•82626 
•82610 
.82593 


•68088 
•68130 
.68173 
.68215 
•68258 




46870 
46778 
46686 
46595 
46503 


.57715 
.57738 
.57762 
•57786 
•57810 


•81664 
•81647 
•81631 
•81614 
•81597 


•70673 
•70717 
•70760 
•70804 
•70848 


1 


41497 
41409 
41322 
41235 
41148 


45 
44 
43 
42 
41. 


20 

21 
22 
23 
24 


.56401 
.56425 
. 56449 
•56473 
•56497 


.82577 
.82561 
.82544 
.82528 
•82511 


.68301 
.68343 
•68386 
•68429 
.68471 


1 


46411 
46320 
46229 
46137 
46046 


•57833 
57857 
57881 
•57904 
•57928 


.81580 
•81563 
•81546 
•81530 
•81513 


•70891 
.70935 
•70979 
•71023 
.71066 




41061 
40974 
40887 
40800 
40714 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.56521 
•56545 
•56569 
56593 
56617 


•82495 
.82478 
.82462 
.82446 
•82429 


.68514 1 
.68557 i 1 
.68600 ' 1 
•68642 j 1 
•68685 1 1 


45955 
45864 
45773 
45682 
45592 


•57952 
•57976 
•57999 
•58023 
•58047 


•81496 
81479 
•81462 
•81445 
•81428 


•71110 
.71154 
.71198 
.71242 
.71285 




40627 
40540 
40454 
40367 
40281 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.56641 
.56665 
•56689 
•56713 
56736 
•56760 
.56784 
.56808 
.56832 
.56856 


.82413 
•82396 
.82380 
.82363 
•82347 


•68728 
.68771 
.68814 
.68857 
•68900 


1 


45501 
45410 
45320 
45229 
45139 


•58070 
.58094 
•58118 
•58141 
58165 


•81412 
•81395 
•81378 
•81361 
•81344 


•71329 
•71373 
•71417 
•71461 
.71505 


1 


40195 
40109 
40022 
39936 
39850 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.82330 
.82314 
.82297 
•82281 
•82264 


.68942 
68985 
.69028 
.69071 
•69114 




45049 
44958 
44868 
44778 
44688 


•58189 
•58212 
•58236 
•58260 
58283 


•81327 
•81310 
•81293 
•81276 
•81259 


.71549 
•71593 
•71637 
•71681 
•71725 




39764 
39679 
39593 
39507 
39421 


25 
24 
23 

22 
21 


40 

41 
42 
43 
44 


.56880 
.56904 
.56928 
•56952 
•56976 


•82248 
•82231 
•82214 
82198 
82181 


•69157 
•69200 
.69243 
.69286 
69329 




44598 
44508 
44418 
44329 
44239 


•58307 
58330 

•58354 
58378 
58401 


•81242 
•81225 
•81208 
•81191 
•81174 


•71769 
•71813 
•71857 
•71901 
71946 




39336 
39250 
•39165 
•39079 
38994 


20 

19 
18 
17 
16 


45 

46 
47 
48 
49 


•57000 
•57024 
•57047 
.57071 
•57095 


•82165 
•82148 
•82132 
•82115 
•82098 


•69372 
•69416 
•69459 
69502 
69545 




44149 
44060 
43970 
43881 
43792 


•58425 
• 58449 
•58472 
58496 
58519 
58543 
58567 
58590 
58614 
•58637 


•81157 
81140 
81123 
81106 
81089 


71990 
•72034 
.72078 
.72122 
•72167 




•38909 
38824 
38738 
38653 
38568 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.57119 
.57143 
•57167 
•57191 
•57215 


.82082 
.82065 
•82048 
82032 
•82015 


69588 
69631 
•69675 
•69718 
•69761 




43703 
43614 
43525 
43436 
43347 


81072 
• 81055 ■■ 

81038 
•81021 
•81004 


•72211 
•72255 
•72299 
• 72344 
•72388 




38484 
38399 
38314 
38229 
38145 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•57238 
.57262 
.57286 
•57310 
•57334 


•81999 
81982 

•81965 
81949 
81932 


•69804 
•69847 
•69891 
•69934 
•69977 




43258 
43169 
43080 
42992 
42903 


•58661 
•58684 
•58708 
58731 
•58755 


•80987 
•80970 1 
•80953 
•80936 
•80919 


•72432 
.72477 
.72521 
.72565 
.72610 




38060 
37976 
37891 
37807 
37722 


5 
4 
3 
2 
1 


60 


•57358 


81915 


.70021 


1 


42815 


•58779 


•80902 ! 


•72654 1 1 


37638 





"^ 


Cos. 


Sin. 


Cot. 


Tan. 1 


Cos. 


Sin. i 


Cot. 


Tan. 


/ 



55^ 



700 



54" 



WTl I 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
36° 37° 





Sin. 


Cos. 


Tan. 


Cot. 


Sin. ! Cos. 1 


Tan. 


Cot. 


' 




1 

2 
3 
4 


.58779 
.58802 
•58826 
•58849 
•58873 


.80902 
.80885 
.80867 
.80850 
.80833 


.72654 
•72699 
•72743 
.72788 
•72832 


1.37638 
1.37554 
1^37470 
1.37386 
1.37302 


•60182 1 
60205 : 
•60228 1 
•60251 
•60274 1 


79864 
79846 
79829 
79811 
79793 


75355 
• 75401 
•75447 
.75492 
•75538 


1.32704 
1.32624 
1^32544 
1. 32464 
1^ 32384 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


58896 

•58920 

.58943 

58967 

58990 


•80816 
.80799 
.80782 
•80765 
•80748 


•72877 
•72921 
.72966 
.73010 
.73055 
.73100 
.73144 
73189 
.73234 
•73278 


1.37218 
1.37134 
1-37050 
1.36967 
1.36883 


.60298 1 
.60321 
.60344 
•60367 
•60390 1 


.79778 
.79758 
•79741 
•79723 
•79706 


■75584 
•75629 
•75675 
.75721 
•75767 


1^32304 
1^32224 
1^32144 
1-32064 
1^ 31984 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


•59014 
.59037 
•59061 
59084 
.59108 


•80730 
•80713 
•80696 
•80679 
•80662 


1.36800 
1.36716 
1.36633 
1.36549 
1.36466 


•60414 1 
•60437 1 
. 60460 
. 60483 
60506 


•79688 
•79671 
•79653 
.79635 
.79618 


•75812 
•75858 
.75904 
•75950 
•75996 


1-31904 
1-31825 
1^31745 
1-31666 
1-31586 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.59131 
•59154 
59178 
•59201 
•59225 


•80644 
•80627 
.80610 
•80593 
.80576 


.73323 
73368 
•73413 
•73457 
•73502 


1.36383 
1.36300 
1.36217 
1-36134 
1-3605] 


•60529 
•60553 
•60576 
•60599 
•60622 


•79600 
•79583 
•79565 
.79547 
.79530 


•76042 
.76088 

76134 
•76180 

76226 


1-31507 
1-31427 
1-31348 
1-31269 
1-31190 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


•59248 
•59272 
•59295 
•59318 
•59342 


•80558 
•80541 
.80524 
.80507 
•80489 


•73547 
•73592 
•73637 
.73681 
.73726 


1.35968 
1.35885 
1.35802 
1.35719 
1.35637 


•60645 
•60668 
•60691 
•60714 
•60738 
• 60761 
•60784 
•60807 
•60830 
•60853 


79512 
.79494 
.79477 
.79459 
• 79441 


•76272 
-76318 
.76364 
.76410 
.76456 


1-31110 
1.31031 
1^30952 
1-30873 
1. 30795 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.59365 
.59389 
.59412 
•59436 
.59459 


•80472 
.80455 
.80438 
.80420 
. 80403 


.73771 
•73816 
•73861 
•73906 
•73951 


1.35554 
1.35472 
1.35389 
1.35307 
1.35224 


•79424 
• 79406 
79388 
•79371 
.79353 


.76502 
.76548 
.76594 
•76640 
•76686 


r 30716 
1-30637 
1- 30558 
1 . 30480 
1 - 30401 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


•59482 
.59506 
•59529 
•59552 
.59576 


.80386 
•80368 
•80351 
• 80334 
•80316 


.73996 
.74041 
. 74086 
.74131 
.74176 


1.35142 
1.35060 
1.34978 
1^34896 
1.34814 


. 60876 
.60899 
60922 
.60945 
.60968 


.79335 
.79318 
.79300 
.79282 
•79264 


.76733 
.76779 
.76825 
.76871 
.76918 


1-30323 
1^30244 
1. 30166 
1-30087 
1-30009 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.59599 
.59622 
.59646 
•59669 
•59693 


•80299 
•80282 
.80264 
.80247 
•80230 


74221 
•74267 
•74312 
.74357 
. 74402 


1^ 34732 
1^34650 
1^34568 
!• 34487 
1-34405 


.60991 
.61015 
.61038 
61061 
•61084 


79247 
.79229 
.79211 
•79193 
•79176 


.76964 
•77010 
.77057 
.77103 
.77149 


1-29931 
1^29853 
1^29775 
1.29696 
1.29£18 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•59716 
•59739 
•59763 
.59786 
•59809 


.80212 
.80195 
.80178 
.80160 
•80143 


. 74447 
•74492 
.74538 
.74583 
.74628 


1.34323 
1.34242 
1.34160 
1.34079 
1.33998 


•61107 
•61130 
•61153 
•S1176 
•61199 


•79158 
•79140 
•79122 
•79105 
•79087 


.77196 
.77242 
•77289 
.77335 
.77382 


1^29541 
1^29463 
1^29385 
1.29307 
1^29229 


30 

19 
18 
17 
16 


45 
46 
47 
48 
40 


.59832 
•59856 
.59879 
.59902 
•59926 


.80125 
.80108 
.80091 
.80073 
.80056 
.80038 
.80021 
•80003 
.79986 
•79968 


.74674 
.74719 
.74764 
.74810 
.74855 


1.33916 
1.33835 
1.33754 
1.33673 
1.33592 


.61222 
.61245 
.61268 
.61291 
.61314 


.79069 
.79051 
•79033 
•79016 
•78998 


•77428 
•77475 
•77521 
•77568 
•77615 


1^29152 
1-29074 
1-28997 
1-28919 
1-28842 


15 
14 
13 
12 
11 


50 

51 

52 
53 

54 


.59949 
.59972 
.59995 
.60019 
•60042 


.74900 
.74946 
.74991 
.75037 
.75082 


1.33511 
1.33430 
1.33349 
1.33268 
1.33187 


•61337 
.61360 
.61383 
.61406 
•61429 


•78980 
•78962 
•78944 
.78926 
•78908 


•77661 
•77708 
.77754 
.77801 
.77848 


1.28764 
1.28687 
1.28610 
1.28533 
1.28456 


10 

9 
8 

7 
6 


55 
56 
57 
58 

59 


•60065 
.60089 
•60112 
•60135 
•60158 
.60182 


.79951 
.79934 
.79916 
.79899 
•79881 


.75128 
.75173 
.75219 
.75264 
.75310 


1.33107 
1.33026 
1.32946 
1.32865 
1.32785 


•61451 
•61474 
•61497 
•61520 
•61543 


•78891 
.78873 
•78855 
.78837 
78819 


.77895 
.77941 
•77988 
•78035 
.78082 


1-28379 
1.28302 
1.28225 
1.28148 
1-28071 


5 
4 
3 

2 

1 


60 


.79864 


.75355 


1.32704 


.61566 


.78801 


•78129 


1-27994 





' 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


' 



63° 



701 



53° 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
38° 39° 



/ 


Sin. 


Cos. 


Tan. 


Cot. 1 


Sin. 


Cos. 


Tan. 


Cot. 


' 




1 

2 
3 

4 


•61566 
.61589 
.61612 
.61635 
.61658 


•78801 
.78783 
•78765 
•78747 
.78729 


•78129 
•78175 
•78222 
•78269 
•78316 




27994 
27917 
27841 
27764 
27688 


•62932 
.62955 
.62977 
63000 
•63022 


•77715 
•77696 
•77678 
•77660 
•77641 


•80978 
•81027 
•81075 
.81123 
.81171 


1.23490 
1.23416 
1.23343 
1.23270 
1.23196 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


•61681 
•61704 
•61726 
•61749 
61772 


•78711 
.78694 
•78676 
•78658 
• 78640 


.78363 
.78410 
•78457 
•78504 
•78551 


1 


27611 
27535 
27458 
27382 
27306 


•63045 
•63068 
•63090 
•63113 
•63135 

•63158 
63180 
63203 
63225 

•63248 


•77623 
•77605 
77586 
•77568 
.77550 


•81220 
•81268 
•81316 
•81364 
•81413 


1-23123 
1.23050 
1.22977 
1.22904 
1.22831 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


•61795 
•61818 
•61841 
•61864 
61887 


•78622 
.78604 
.78586 
•78568 
•78550 


•78598 
•78645 
•78692 
.78739 
78786 


1 


27230 
27153 
27077 
27001 
26925 


.77531 
.77513 
•77494 
.77476 
•77458 


•81461 
•81510 
•81558 
•81606 
•81655 


1-22758 
1.22685 
1-22612 
1-22539 
1-22467 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


•61909 
•61932 

61955 
•61978 

62001 


•78532 
•78514 
•78496 
78478 
•78460 


78834 
78881 
•78928 
•78975 
•79022 




26849 
26774 
26698 
26622 
26546 


•63271 
•63293 
63316 
•63338 
•63361 


•77439 
•77421 
•77402 
•77384 
•77366 


.81703 
.81752 
.81800 
•81849 
•81898 


1.22394 
1.22321 
1.22249 
1.22176 
1-22104 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


•62024 
•62046 
•62069 
62092 
62115 


• 78442 
•78424 
•78405 
•78387 
•78369 


•79070 
•79117 
•79164 
•79212 
79259 




26471 
26395 
26319 
26244 
26169 


63383 
63406 
63428 
•63451 
63473 


•77347 
.77329 
•77310 
•77292 
.77273 


■81946 
•81995 
■82044 
•82092 
.82141 


1-22031 
1-21959 
1-21886 
1-21814 
1-21742 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•62138 
•62160 
■62183 
•62206 
•62229 

•62251 
.62274 
.62297 
.62320 
•62342 


•78351 
78333 
•78315 
•78297 
•78279 


•79306 
.79354 
•79401 
. 79449 
.79496 




26093 
26018 
25943 
25867 
25792 


63496 
•63518 
•63540 
•63563 
•63585 


.77255 
.77236 
•77218 
■77199 
■77181 


.82190 
.82238 
•82287 
•82336 
•82385 


1-21670 
1-21598 
1-21526 
1-21454 
1-21382 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.78261 
•78243 
•78225 
•78206 
78188 


•79544 1 
•79591 1 
•79639 i 1 
•79686 1 1 
•79734 1 1 


25717 
25642 
25567 
25492 
25417 


•63608 
•63630 
•63653 
•63675 
•63698 


•77162 
•77144 
•77125 
•77107 
•77088 


■82434 
82483 
■82531 
■82580 
■82629 


1-21310 
1-21238 
1-21166 
1-21094 
1^21023 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.62365 
•62388 
.62411 
•62433 
•62456 

•62479 

• 62502 
•62524 

• 62547 
•62570 


■78170 
•78152 
•78134 
•78116 
•78098 

•78079 
.78061 
.78043 
.78025 
.78007 


.79781 
.79829 
.79877 
.79924 
•79972 
•80020 
•80067 
•80115 
.80163 
•80211 


1 


25343 
25268 
25193 
25118 
250^^ 
24969 
24895 
24820 
•24746 
.24672 


63720 
•63742 
•63765 
•63787 
•63810 

•63832 
•63854 
63877 
■63899 
•63922 


.77070 
•77051 
■77033 
■77014 
•76996 


•82678 
■82727 
•82776 
•82825 
.82874 


1-20951 
1^20879 
1-20808 
1-20736 
1-20665 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•76977 
•76959 
■76940 
•76921 
■76903 


•82923 
•82972 
•83022 
•83071 
•83120 


1-20593 
1-20522 
1-20451 
1-20379 
1-20308 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


• 62592 
•62615 
•62638 
•62660 
•62683 


•77988 
.77970 
•77952 
.77934 
.77916 


•80258 
•80306 
•80354 
•80402 
•80450 




.24597 
.24523 
24449 
24375 
24301 


-63944 
•63966 
•63989 
•64011 
■64033 


76884 
■76866 
•76847 
•76828 
•76810 


83169 
83218 
■83268 
■83317 
■83366 


1-20237 
1-20166 
1.20095 
1^20024 
1.19953 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•62706 
62728 
•62751 
.62774 
•62796 


.77897 
•77879 
•77861 
.77843 
.77824 


80498 
•80546 
•80594 
•80642 
•80690 




24227 
24153 
24079 
24005 
23931 


■64056 
64078 
■64100 
■64123 
•64145 


•76791 
•76772 
.76754 
•76735 
•76717 


■83415 
■83465 
•83514 
•83564 
■83613 


1-19882 
1-19811 
1. 19740 
1- 19669 
1^19599 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•62819 
•62842 
.62864 
.62887 
•62909 


•77806 
77788 
•77769 
•77751 
•77733 


•80738 
•80786 
•80834 
•80882 
.80930 




23858 
23784 
23710 
23637 
23563 


•64167 
■64190 
•64212 
•64234 
64256 


.76698 
•76679 
•76661 
•76642 
•76623 


-83662 
■83712 
■83761 
•83811 
•83860 


1-19528 
1.19457 
1.19387 
1-19316 
1-19246 


5 
4 
3 

2 

1 


60 


•62932 


•77715 


80978 


1 


23490 


•64279 


•76604 


•83910 


1-19175 





' 


Cos. 


Sin. 


Cot. 


Tan. 1 


Cos. 


Sin. 


Cot. 


Tan. 


' 



5V 



702 



50° 



TABLE IX.- 


-NATURAL SINES, COSINES. TANGENTS, AND COTANGENTS. 
40° 41° 


/ 


Sin. 


Cos. 


Tan. Cot. | 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 
2 
3 
4 


. 64279 
•64301 
.64323 
. 64346 
.64368 


.76604 
.76586 
•76567 
76548 
•76530 


•83910 
•83960 
•84009 
.84059 
•84108 


1^19175 
1.19105 
1.19035 
1.18964 
1.18894 


65606 
65628 
65650 
65672 
.65694 


75471 

75452 

.75433 

.75414 

•75395 


86929 
86980 
■87031 
■87082 
■87133 


1.15037 
1.14969 
1-14902 
1.14834 
1.14767 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.64390 
.64412 
.64435 
•64457 
. 64479 


•76511 
•76492 
•76473 
•76455 
.76436 


•84158 
■84208 
•84258 
•84307 
.84357 


1.18824 
1.18754 
1.18684 
1.18614 
1.18544 


•65716 
•65738 
•65759 
■65781 
■65803 


.75375 
.75356 
.75337 
.75318 
.75299 


■87184 
■87236 
•87287 
■87338 
•87389 


1.14699 
1.14632 
1.14565 
1.14498 
1 . 14430 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 


•64501 
•64524 
• 64546 
64568 
•64590 


.76417 
.76398 
•76380 
•76361 
•76342 


■ 84407 
■84457 
•84507 
•84556 
84606 


1.18474 
1.18404 
1.18334 
1.18264 
1.18194 


.65825 
■65847 
■65869 
•65891 
•65913 


.75280 
.75261 
.75241 
.75222 
.75203 


•87441 
■87492 
■87543 
■87595 
■87646 


1.14363 
1.14296 
1.14229 
1.14162 
1-14095 


50 

49 
48 
47 
46 


.64612 
•64635 
•64657 
•64679 
•64701 


.76323 
•76304 
.76286 
•76267 
•76248 


.84656 
.84706 
•84756 
84806 
.84856 


1.18125 
1.18055 
1.17986 
1.17916 
1.17846 


.65935 
■65956 
■65978 
• 66000 
•66022 


.75184 
.75165 
.75146 
•75126 
•75107 


.87698 
.87749 
.87801 
.87852 
.87904 


1-14028 
1-13961 
1-13894 
1.13828 
1-13761 


45 
44 
43 
42 
41 


.64723 
. 64746 
•64768 
•64790 
•64812 


.76229 
.76210 
•76192 
•76173 
•76154 


■84906 
•84956 
.85006 
.85057 
85107 


1.17777 
1.17708 
1.17638 
1.17569 
1.17500 


■ 66044 
•66066 
■66088 
■66109 
■66131 


75088 
•75069 
■75050 
•75030 
■75011 


■87955 
.88007 
.88059 
■88110 
■88162 


1-13694 
1.13627 
1.13561 
1.13494 
1.13428 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•64834 
•64856 
•64878 
•64901 
•64923 


•76135 
•76116 
•76097 
•76078 
.76059 


■85157 
■85207 
■85257 
85308 
■85358 


1.17430 
1.17361 
1.17292 
1.17223 
1.17154 


■66053 
■66175 
■66197 
•66218 
• 66240 


.74992 
.74973 
•74953 
. 74934 
•74915 


.88204 
■88265 
.88317 
.88369 
■88421 


1.13361 
1-13295 
1-1322& 
1-13162 
1-13096 


35 

34 
33 
32 
31 


30 

31 
32 
33 
34 


•64945 
•64967 

64989 
•65011 

65033 


•76041 
•76022 
.76003 
•75984 
.75965 


■85408 
•85458 
•85509 
■85559 
■85609 


1.17085 
1^17016 
1^16947 
1.16878 
1.16809 


.66262 
.66284 
•66306 
.66327 
.66349 


■74896 
■74876 
■74857 
■74838 
■74818 


.88473 
.88524 
■88576 
■88628 
■88680 


1-13029 
1.12963 
1.12897 
1-12831 
1-12765 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


•65055 
•65077 
•65100 
•65122 
•65144 


•75946 
.75927 
•75908 
•75889 
.75870 


■85660 
■85710 
■85761 
•85811 
.85862 


1^16741 
1^16672 
1^16603 
1-16535 
1-16466 


.66371 
•66393 
■66414 
■66436 
•66458 


•74799 
•74780 

• 74760 
■ 74741 

• 74722 


■88732 
■88784 
•88836 
■88888 
•88940 


1-12699 
1-12633 
1-12567 
1-12501 
1-12435 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•65166 
•65188 
•65210 
•65232 
•65254 


•75851 
75832 

•75813 
75794 
75775 


•85912 
•85963 
•86014 
■86064 
■86115 


1.16398 
1.16329 
1.16261 
1.16192 
1.16124 


■66480 
■66501 
■66523 
■66545 
■66566 


• 74703 
- 74683 
■ 74664 
•74644 
■74625 


•88992 
.89045 
•89097 
•89149 
•89201 


1.12369 
1.12303 
1-12238 
1-12172 
1.12106 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


65276 
65298 
•65320 
•65342 
•65364 


•75756 

•75738 

■75719 

75700 

75680 


■86166 
86216 
■86267 
■86318 
86368 


1.16056 
1^15987 
1.15919 
1.15851 
1-15783 


■66588 
•66610 
•66632 
•66653 
•66675 


-74606 
74586 
•74567 

■ 74548 

■ 74528 


■89253 
■89306 
•89358 
•89410 
•89463 


1.12041 
1.11975 
1.11909 
1.11844 
1-11778 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•65386 
.65408 
•65430 
.65452 
.65474 


.75661 
•75642 
•75623 
.75604 
.75585 


86419 
.86470 
■86521 
•86572 
•86623 


1.15715 
1.15647 
1.15579 
1.15511 
1.15443 


•66697 
•66718 
• 66740 
■66762 
■66783 


■ 74509 

■ 74489 
• 74470 
■74451 
•74431 


•89515 
■89567 
■89620 
■89672 
■89725 


1-11713 
1-11648 
1.11582 
1.11517 
1.11452 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 
60 


•65496 
.65518 
.65540 
.65562 
•65584 


•75566 
•75547 
•75528 
•75509 
.75490 


■86674 
■86725 
•86776 
•86827 
•86878 


1.15375 
1.15308 
1.15240 
1^15172 
1.15104 


•66805 
•66827 
•66848 
66870 
■66891 


■74412 
■74392 
•74373 
•74353 
■74334 


■89777 
■89830 
89883 
■89935 
■89988 


1.11387 
1.11321 
1-11256 
1.11191 
1.11126 


5 
4 
3 

2 

1 


.65606 


.75471 


■86929 


1.15037 


■66913 


■74314 


■90040 


1.11061 





/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 


49^ 703 48° 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS^! 
43° 43° 



n 




TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS 
44° 44° 



/ 


Sin. 


Cos. 


Tan. 


Cot. 


/ 


f 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 

4 


•69466 
.69487 
•69508 
.69529 
.69549 


•71934 
•71914 
•71894 
•71873 
71853 


•96569 
•96625 
.96681 
.96738 
•96794 


1-03553 
1^03493 
1^ 03433 
1^03372 
1.03312 


60 

59 
58 

57 
56 


30 

31 

32 
33 
34 


.70091 
.70112 
-70132 
.70153 
.70174 


.71325 
.71305 
.71284 
.71264 
•71243 


.98270 
•98327 
•98384 
.98441 
•98499 


1.01761 
1.01702 
1.01642 
1.01583 
1.01524 


30 

29 
28 
27 

26 


5 
6 
7 

8 
9 


•69570 
.69591 
•69612 
•69633 
•69654 


•71833 
.71813 
.71792 
.71772 
•71752 


•96850 
•96907 
.96963 
.97020 
.97076 


1.03252 
1.03192 
1^03132 
1^03072 
1^03012 


55 
54 
53 
52 
51 


35 
36 
37 
38 
39 


.70195 
.70215 
.70236 
.70257 
•70277 


•71223 
•71203 
•71182 
.71162 
•71141 


.98556 
•98613 
.98671 
•98728 
•98786 


1.01465 
1.01406 
1.01347 
1.01288 
1.01229 


25 
24 
23 
22 
21 


10 

11 
12 
13 
14 


•69675 
•69696 
•69717 
•69737 
•69758 


.71732 
•71711 
•71691 
.71671 
.71650 


.97133 
.97189 
.97246 
.97302 
.97359 


1 02952 
1.02892 
1.02832 
1.02772 
1.02713 


50 

49 
48 
47 
46 


40 

41 
42 
43 
44 


.70298 
•70319 
•70339 
•70360 
•70381 


.71121 
.71100 
.71080 
.71059 
.71039 


98843 
•98901 
•98958 
.99016 
-99073 


1.01170 
1.01112 
1.01053 
1.00994 
1.00935 


20 

19 
18 
17 
16 


15 
16 
17 

18 
19 
20 

21 
22 
23 
24 

25 
26 
27 
28 
29 
3^ 


•69779 
•69800 
•69821 
.69842 
.69862 


.71630 
•71610 
•71590 
•71569 
•71549 


.97416 
.97472 
.97529 
•97586 
.97643 


1.02653 
1.02593 
1.02533 
1.02474 
i. 02414 


45 
44 
43 
42 
41 


45 
46 
47 
43 
49 


• 70401 
. 70422 
70443 
.70463 
. 70484 


.71019 
.70998 
.70978 
.70957 
70937 


-99131 
-99189 
.99247 
.99304 
•99362 


1.00876 
1.00818 
1^00759 
1.00701 
1.00642 


15 
14 
13 
12 
11 


•69883 
•69904 
•69925 
•69946 
•69966 


•71529 
•71508 
•71488 
•71468 
•71447 


•97700 
.97756 
.97813 
.97870 
•97927 


1.02355 
1-02295 
1.02236 
1.02176 
1.02117 


40 

39 
38 
37 
36 


50 

51 
52 
53 
54 


-70505 
.70525 
. 70546 
.70567 
• 70587 


.70916 
.70896 
•70875 
•70855 
•70834 
•70813 
•70793 
•70772 
• 70752 
•70731 


•99420 
•99478 
•99536 
•99594 
•99652 


1^ 00583 
1^00525 
1.00467 
1.00408 
1.00350 


10 
9 

8 
7 

6 


69987 

• 70008 
•70029 
. 70049 

• 70070 


.71427 
.71407 
.71386 
.71366 
.71345 


•97984 
98041 
.98098 
.98155 
•98213 
-98270 


1.02057 
1.01998 
1.01939 
1.01879 
1.01820 


35 
34 
33 
32 
31 


55 
56 
57 
58 
59 


• 70608 
.70628 
.70649 
- 70670 
. 70690 


.99710 
.99768 
.99826 
.99884 
.99942 


1.00291 
1.00233 
1.00175 
1.00116 
1.00058 


5 

4 
3 

2 
1 


•70091 

Cos. 


•71325 


1.01761 


30 


60 


•70711 


•70711 


1 • OOOOO 


1 . OOOOO 





Sin. 


Cot. 


Tan. 


' 


' 


Cos. 


Sin. 


Cot. 


Tan. 


^ 








45° 




7 


05 




45^ 









TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
0° 1° 2° 3° 



' 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


' 




1 

2 
3 
4 


.00000 
.00000 
.00000 
.00000 
. 00000 




00000 
00000 
00000 
00000 
00000 


•00015 
•00016 
.00016 
•00017 
•00017 




00015 
00016 
00016 
00017 
00017 


.00061 
•00062 
•00063 
•00064 
•00065 




00061 
00062 
00063 
00064 
00065 
00066 
00067 
00068 
00069 
00070 


•00137 
•00139 
•00140 
•00142 
■00143 




00137 
00139 
00140 
00142 
00143 




1 
2 
3 
4 


5 
6 
7 
8 
9 


•00000 
.00000 
.00000 
.00000 
.00000 




00000 
00000 
00000 
00000 
00000 


•00018 
•00018 
•00019 
.00020 
.00020 




00018 
00018 
00019 
00020 
00020 


.00066 
.00067 
.00068 
.00069 
•00070 


■00145 
00146 
•00148 
•00150 
•00151 




00145 
00147 
00148 
00150 
00151 


5 
6 
• 7 
8 
9 


10 

11 
12 
13 
14 


.00000 
.00001 
.00001 
.00001 
.00001 




00000 
00001 
00001 
00001 
00001 


.00021 
.00021 
•00022 
•00023 
•00023 




00021 
00021 
00022 
00023 
.00023 


•00071 
•00073 
•00074 
00075 
00076 




00072 
.00073 
.00074 
•00075 
•00076 


•00153 
•00154 
•00156 
•00158 
•00159 




00153 
00155 
•00156 
•00158 
•00159 


10 

11 

12 
13 
14 


15 
16 
17 
18 
19 


.00001 
.00001 
.00001 
00001 
.00002 




00001 
00001 
00001 
00001 
00002 


•00024 
•00024 
•00025 
•00026 
00026 




.00024 
00024 
00025 
00026 
00026 


•00077 
•00078 
•00079 
00081 
■00082 




.00077 
•00078 
•00079 
•00081 
■00082 


•00161 
•00162 
•00164 
•00166 
•00168 




•00161 
.00163 
.00164 
.00166 
.00168 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


.00002 
. 00002 
00002 
00002 
00002 




00002 
00002 
00002 
00002 
00002 


•00027 
•00028 
•00028 
•00029 
•00030 




00027 
00028 
00028 
00029 
00030 


■00083 
00084 
00085 
-00087 
■00088 




00083 
.00084 
•00085 
•00087 

00088 


00169 
•00171 
•00173 
•00174 
■00176 




•00169 
•00171 
•00173 
•00175 
•00176 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


00003 
00003 
00003 
00003 
00004 




.00003 
.00003 
■00003 
•00003 
00004 


•00031 
•00031 
•00032 
•00033 
00034 




00031 
00031 
00032 
00033 
00034 


• 00089 
00090 
•00091 
•00093 
•00094 




•00089 
•00090 
•00091 
00093 
00094 


■0017o 
•00179 
•00181 
•00183 
•00185 




•00178 
.00180 
•00182 
•00183 
.00185 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•00004 
. 00004 
• 00004 
00005 
00005 




.00004 
.00004 
00004 
00005 
00005 


•00034 
•00035 
00036 
•00037 
•00037 




00034 
00035 
00036 
00037 
00037 


00095 
•00096 
■00098 
■00099 
•00100 




000S5 
00097 
00098 
00099 
00100 


00187 
00188 

■00190 
00192 

•00194 




.00187 
.00189 
.00190 
•00192 
•00194 


30 

31 
32 
33 
34 


35 
36 
37 
38 
,39 


00005 
00005 
•00006 
00006 
00006 




00005 
00005 
00006 
00006 
00006 


•00038 
•00039 
.00040 
.00041 
00041 




00038 
00039 
00040 
00041 
00041 


00102 
•00103 
■00104 
00106 
00107 




00102 
.00103 
•00104 
00106 
00107 


■00196 
■00197 
■00199 
■00201 
■00203 




•00196 
.00198 
.00200 
.00201 
00203 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


00007 
00007 
•00007 
00008 
00008 




•00007 
•00007 

00007 
.00008 

00008 


•00042 
00043 
00044 
•00045 
■00046 




00042 
00043 
00044 
00045 
00046 


■00108 
00110 
.00111 
■00112 
■00114 




00108 
00110 
00111 
00113 
00114 


■00205 
■00207 
■00208 
■00210 
00212 




00205 
00207 
00209 
00211 
00213 


40 

41 
42 
43 
44 


45 
46 
47 
47 
49 


00009 
00009 
00009 
•00010 
00010 




00009 
00009 
00009 
00010 
00010 


■00047 
00048 

■00048 
00049 

■00050 




00047 
00048 
00048 
00049 
00050 


■00115 
■ 00U7 
■00118 
■00119 
■00121 




00115 
00117 
00118 
00120 
00121 


■00214 
00216 
■00218 
■00220 
■00222 




00215 
^0216 
00218 
0U220 
00222 


' 45 
46 
47 
48 
49 


60 

51 
52 
53 
54 


00011 
•00011 
•00011 

00012 
•00012 




00011 
00011 
00011 
00012 
00012 


•00051 
•00052 
•00053 
•00054 
•00055 




00051 
00052 
00053 
00054 
00055 


•00122 
•00124 
•00125 
■00127 
•00128 




00122 
00124 
00125 
00127 
00128 


■00224 
•00226 
■00228 
00230 
00232 




00224 
00226 
00228 
00230 
00232 


oO 

51 
52 
53 
54 


55 
56 
57 
58 
59 


00013 
•00013 
00014 
00014 
00015 




.00013 
00013 

.00014 
00014 
00015 


00056 
•00057 
■00058 
•00059 

00060 




00056 
00057 
00058 
00059 
00060 


00130 
•00131 
■00133 
•00134 

00136 




00130 
00131 
00133 
00134 
00136 


■00234 
00236 
00238 
00240 

■00242 




00234 
00236 
C0238 
00240 
00242 


55 
56 
57 
58 
59 


60 


•00015 




00015 


•00061 


•00061 


00137 




00137 


•00244 j . 


00244 


60 



706 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL 
4° 5° 6° 7° 


SECANTS. 


' 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


' 




1 
2 
3 
4 


00244 
00246 
00248 
00250 
•00252 




00244 
00246 
00248 
00250 
00252 


.00381 
00383 
•00386 
•00388 
•00391 




00382 
00385 
00387 
00390 
00392 


00548 
00551 
.00554 
.00557 
.00560 




00551 
00554 
00557 
00560 
00563 


•00745 
.00749 

00752 
•00756 

00760 




00751 
00755 
00758 
00762 
00765 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.00254 
.00256 
.00258 
.00260 
.00262 




00254 
00257 
00259 
00261 
00263 


.00393 
.00396 
.00398 
.00401 
•00404 
.00406 
.00409 
.00412 
.00114 
.00417 
.00420 
.00422 
.00425 
.00428 
.00430 




00395 
00397 
00400 
00403 
00405 


.00563 
.00566 
.00569 
.00572 
.00576 




00566 
00569 
00573 
00576 
00579 


•00763 
•00767 
00770 
.00774 
.00778 




00769 
00773 
00776 
00780 
00784 


5 
6 
7 
8 
9 


10 

11 
12 
]3 
14 


00264 
.00266 

00269 
.00271 

00273 




00265 
00267 
00269 
00271 
00274 




00408 
00411 
00413 
00416 
00419 


00579 
.00582 
00585 
00588 
.00591 




00582 
00585 
00588 
00592 
00595 


.00781 
.00785 
•00789 
00792 
•00796 




00787 
00791 
00795 
00799 
00802 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 
20 
21 
22 
23 
24 


00275 
.00277 

00279 
.00281 
•00284 




00276 
00278 
00280 
00282 
00284 




00421 
00424 
00427 
00429 
00432 


00594 
00598 
.00601 
.00604 
.00607 




00598 
00601 
00604 
00608 
00611 


• C0800 
•00803 
•00807 
•00811 
•00814 




00806 
00810 
00813 
00817 
00821 


15 
16 
17 
18 
19 


.00286 
00288 
.00290 
.00293 
00295 




00287 
00289 
00291 
00293 
00296 


.00433 
.00436 
.00438 
.00441 
00444 




00435 
00438 
00440 
00443 
00446 


•00610 
00614 
•00617 
•00620 
■00623 




00614 
00617 
00621 
00624 
00627 


.00818 
.00822 
.00825 
00829 
•00833 




00825 
00828 
00832 
00836 
00840 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


00297 
00299 
.00301 
.00304 
00306 




00298 
00300 
00302 
00305 
00307 


.00447 
. 00449 
.00452 
.00455 
.00458 




00449 
00451 
00454 
00457 
00460 


.00626 
00630 
00633 
00636 
00640 




00630 
00634 
00637 
00640 
00644 


.00837 
00840 
.00844 
.00848 
.00852 




00844 
00848 
00851 
00855 
00859 


25 
26 
27 
28 
29 


39 

31 
32 
33 
34 


.00308 
.00311 
.00313 
00315 
.00317 




00309 
00312 
00314 
00316 
00318 


.00460 
•00463 
00466 
.00469 
.00472 




00463 
00465 
00468 
00471 
00474 


.00643 
•00646 
•00649 
•00653 
•00656 




00647 
00650 
00654 
00657 
00660 


.00856 
00859 

-00863 
00867 

.00871 




00863 
00867 
00871 
00875 
00878 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


00320 
• 00322 
.00324 
00327 
00329 




00321 
•00323 
•00326 
•00328 
•00330 


•00474 
.00477 
•00480 
00483 
■00486 




00477 
00480 
.00482 
.00485 
00488 


•00659 
•00663 
•00666 
•00669 
00673 




00664 
•00667 
•00671 
.00674 
•00677 


00875 
•00878 

00882 
•00886 
•00890 




00882 
00886 
00890 
00894 
00898 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
5i 
52 
53 
54 


.00332 
.00334 
00336 
• 00339 
.00341 




•00333 
•00335 
.00337 
00340 
.00342 


•00489 
.00492 
•00494 
•00497 
•00500 
•00503 
00506 
00509 
•00512 
•00515 




.00491 
.00494 
.00497 
•00500 
•00503 


•00676 
.00680 

00683 
•00686 

00690 




.00681 
.00684 
.00688 
.00691 
.00695 


.00894 
00898 
.00902 
.00906 
00909 




00902 
00906 
00910 
00914 
00918 


40 

41 
42 
43 
44 


. 00343 
.00346 
.00348 
00351 
00353 




.00345 
.00347 
00350 
.00352 
.00354 




.00506 
.00509 
.00512 
00515 
.00518 


00693 
00607 
.00700 
.00703 
.00707 




.00698 
.00701 
.00705 
■00708 
.00712 


00913 
00917 
00921 
00925 
•00929 




00922 
00926 
00930 
00934 
00938 


45 
46 
47 
48 
49 


.00356 
00358 
00361 
.0036d 
.00365 




•00357 
.00359 
.00362 
.00364 
•00367 


•00518 
.00521 
.00524 
.00527 
•00530 




.00521 
.00524 
.00527 
.00530 
•00533 


.00710 
.00714 
.00717 
.00721 
.00724 




•00715 
.00719 
.00722 
•00726 
•00730 


•00933 
00937 
.00941 
.00945 
■00949 




00942 
00946 
00950 
•00954 
.00958 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 
60 


.00368 
.00370 
.00373 
.00375 
00378 




■00369 
.00372 
•00374 
00377 
•00379 


.00533 
•00536 
.00539 
.00542 
00545 




.00536 
•00539 
.00542 
•00545 
00548 


.00728 
.00731 
.00735 
.00738 
•00742 




•00733 
•00737 
.00740 
.00744 
.00747 


.00953 
•00957 
00961 
00965 
.00969 




.00962 
•00966 
•00970 
•00975 
.00979 


55 
56 
57 
58 
59 


•00381 




•00382 


00548 


•00551 1 


•00745 




.00751 


.00973 




.00983 


60 














7( 


37 















TABLE X— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
8° 9° 10° ll*" 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 

4 


.00973 
.00977 
•00981 
•00985 
•00989 




00983 
00987 
00991 
00995 
00999 


.01231 
•01236 
.01240 
.01245 
.01249 


- 


01247 
01251 
01256 
01261 
01265 
01270 
01275 
01279 
01284 
01289 


•01519 
.01524 
.01529 
.01534 
.01540 




01543 
01548 
01553 
01558 
01564 


•01837 
•01843 
•01848 
.01854 
•01860 




01872 
01877 
01883 
01889 
01895 



1 

2 
3 
4 


5 
6 
7 
8 
9 


00994 
-00998 
.01002 
.01006 
.01010 




01004 
01008 
01012 
01016 
01020 


•01254 
.01259 
.01263 
•01268 
•01272 


.01545 
•01550 
.01555 
.01560 
.01565 




01569 
01574 
01579 
01585 
01590 


.01865 
01871 
•01876 
•01882 
01888 




01901 
01906 
01912 
01918 
01924 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•01014 
.01018 
.01022 
•01027 
•01031 




01024 
01029 
01033 
01037 
01041 


.01277 
•01282 
•01286 
•01291 
•01296 




01294 
01298 
01303 
01308 
01313 


•01570 
•01575 
•01580 
•01586 
•01591 




01595 
01601 
01606 
01611 
01616 


.01893 
.01899 
.01904 
.01910 
•01916 




01930 
01936 
01941 
01947 
01953 


10 

11 
12 
13 
14 


15 
16 
17 
18 

19 


01035 
01039 
01043 
01047 
01052 




01046 
01050 
01054 
01059 
01063 


•01300 
•01305 

01310 
■01314 

01319 




01318 
01322 
01327 
01332 
01337 


•01596 
01601 
•01606 
•01612 
•01617 




01622 
01627 
01633 
01638 
01643 


•01921 
.01927 
.01933 
.01939 
•01944 




01959 
01965 
01971 
01977 
01983 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


01056 
01060 
01064 
•01069 
.01073 




01067 
01071 
01076 
01080 
01084 


•01324 
•01329 
•01333 
•01338 
.01343 




01342 
01346 
01351 
01356 
01361 


.01622 
.01627 
.01632 
.01638 
•01643 




01649 
01654 
01659 
01665 
01670 


•01950 
•01956 
•01961 
•01967 
•01973 




01989 
01995 
02001 
02007 
02013 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.01077 
.01081 
.01086 
•01090 
•01094 




01089 
01093 
01097 
01102 
01106 


.01348 
•01352 
.01357 
.01362 
.01367 

•01371 
•01376 
•01381 
.01386 
.01391 




01366 
01371 
01376 
01381 
01386 


.01648 
01653 
.01659 
•01664 
•01669 

•01675 
.01680 
•01685 
•01690 
•01696 




01676 
01681 
01687 
01692 
01698 


■01979 
.01984 
•01990 
•01996 
•02002 




02019 
02025 
.02031 
.02037 
.02043 


25 
26 
27 
28 

29 


30 

31 
32 
33 
34 


•01098 
•01103 
01107 
•01111 
•01116 


01111 
01115 
01119 
01124 
01128 




01391 
01395 
01400 
01405 
01410 




01703 
01709 
01714 
01720 
01725 


.02008 
.02013 
.02019 
.02025 
•02031 




.02049 
.02055 
•02061 
•02067 
02073 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•01120 
•01124 
•01129 
•01133 
•01137 




01133 
01137 
01142 
01146 
01151 


.01396 
-01400 
•01405 
•01410 
.01415 




01415 
01420 
01425 
01430 
01435 


.01701 
.01706 
.01712 
•01717 
■01723 




01731 
01736 
01742 
01747 
01753 


•02037 
•02042 
•02048 
■02054 
02060 




02079 
.02085 
02091 
02097 
02103 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


01142 
•01146 
•01151 
•01155 
•01159 




01155 
01160 
01164 
01169 
01173 


■01420 
.01425 
•01430 
-01435 
.01439 




01440 
01445 
01450 
01455 
01461 


■01728 
01733 
.01739 
.01744 
•01750 




01758 
01764 
01769 
01775 
01781 


•02066 
•02072 
•02078 
.02084 
-02090 




02110 
02116 
02122 
02128 
02134 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•01164 
•01168 
01173 
•01177 
•01182 




.01178 
01182 
01187 
01191 
01196 


.01444 
•01449 
.01454 
•01459 
.01464 




01466 
01471 
01476 
01481 
01486 


•01755 
.01760 
01766 
•01771 
•01777 




01786 
01792 
01793 
01803 
01809 


.02095 
.02101 
.02107 
.02113 
•02119 




02140 
02146 
02153 
02159 
02165 
•02171 
02178 
02184 
02190 
02196 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


01185 
•01191 
•01195 
•01200 
•01204 




01200 
01205 
01209 
.01214 
01219 


.01469 
•01474 
.01479 
•01484 
•01489 




01491 
01496 
01501 
01506 
01512 


.01782 
.01788 
.01793 
.0179t 
•01804 




01815 
01820 
01826 
01832 
01837 


•02125 
.02131 
•02137 
•02143 
.02149 


50 

51 
52 
53 
54 


55 
56 
57 
58 
5L 


•01209 
.01213 
•01218 
•01222 
•01227 




.01223 
01228 
.01233 
.01237 
01242 


.01494 
.01499 
.01504 
.01509 
01514 




01517 
01522 
01527 
01532 
01537 


•01810 
.01815 
.01821 
.01826 
•01832 




01843 
01849 
01854 
01860 
01866 


•02155 
.02161 
•02167 
•02173 
•02179 
02185 


- 


02203 
02209 
02215 
02221 
02228 
02234 


55 
56 
57 
58 
59 


60 


01231 




.01247 


.01519 




01543 


•01837 




01872 


60 



708 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
13° 13° 14° 15° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.02185 
.02191 
02197 
.02203 
.02210 


.02234 
.02240 
.02247 
.02253 
.02259 


02563 
02570 
02576 
02583 
02589 


.02630 
.02637 
.02644 
.02651 
.02658 


.02970 
.02977 
.02985 
.02992 
.02999 


•03061 
•03069 
.03076 
•03084 
•03091 


03407 
.03415 
.03422 
.03430 
•03438 


.03528 
.03536 
.03544 
.03552 
-03560 




1 
2 
3 
4 


5 
6 
7 
8 

9 


.02216 
.02222 
.02228 
.02234 
.02240 


.02266 
.02272 
.02279 
.02285 
.02291 


02596 
•02602 
.02609 
.02616 
•02622 


.02665 
.02672 
.02679 
.02686 
.02693 


.03006 
.03013 
.03020 
•03027 
.03034 

.03041 
•03048 
•03055 
.03063 
•03070 


.03099 
.03106 
.03114 
.03121 
.03129 


•03445 
•03453 
03460 
•03468 
.03476 


.03568 
.03576 
.03584 
.03592 
-03601 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.02246 
.02252 
.02258 
.02265 
.02271 


.02298 
02304 
.02311 
•02317 
.02323 


•02629 
•02635 
.02642 
.02649 
.02655 


.02700 
.02707 
-02714 
.02721 
.02728 


.03137 
.03144 
.03152 
.03159 
•03167 


•03483 
•03491 
•03498 
•03506 
•03514 


.03609 
.03617 
.03625 
.03633 
.03642 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


02277 
.02283 
•02289 
.02295 
.02302 


.02330 
.02336 
.02343 
.02349 
.02356 


.02662 
.02669 
.02675 
•02682 
.02689 


.02735 
.02742 
.02749 
•02756 
.02763 


.03077 
.03084 
•03091 
•03098 
.03106 


•03175 
•03182 
•03190 
•03198 
•03205 


•03521 
.03529 
.03537 
.03544 
.03552 


.03650 
.03658 
.03666 
.03674 
•03683 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


.02308 
.02314 
.02320 
.02327 
.02333 


.02362 
.02369 
.02375 
.02382 
•02388 


.02696 
•02702 
•02709 
.02716 
.02722 


.02770 
.02777 
.02784 
.02791 
.02799 


.03113 
.03120 
.03127 
-031S4 
.03142 


•03213 
•03221 
•03228 
.03236 
.03244 


.03560 
•03567 
03575 
.03583 
.03590 


•03691 
.03699 
.03708 
.03716 
.03724 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.02339 
.02345 
•02352 
02358 
.02364 


.02395 
.02402 
.02408 
•02415 
.02421 


.02729 
.02736 
.02743 
.02749 
•02756 


.02806 
.02813 
.02820 
.02827 
.02834 


•03149 
.03156 
•03163 
03171 
•03178 


.03251 
.03259 
.03267 
.03275 
•03282 


.03598 
.03606 
.03614 
.03621 
.03629 


•03732 
•03741 
.03749 
.03758 
•03766 


2§ 
26 
27 
28 
29 


30 

31 
32 
33 
34 


02370 
•02377 
.02383 
.02389 

02396 


•02428 
.02435 
•02441 
•02448 
•02454 


.02763 
.02770 
•02777 
•02783 
•02790 


.02842 
.02849 
.02856 
.02863 
.02870 


-03185 
.03193 
.03200 
.03207 
.03214 


.03290 
•03298 
•03306 
•03313 
.03321 


•03637 
•03645 
.03653 
.03660 
.03668 


•03774 
.03783 
•03791 
.03799 
•03808 
•03816 
•03825 
•03833 
•03842 
•03850 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•02402 
02408 
•02415 
-02421 
•02427 


.02461 
•02468 
•02474 
-02481 
.02488 


•02797 
•02804 
•02811 
•02818 
02824 


.02878 
.02885 
.02892 
•02899 
.02907 


.03222 
.03229 
.03236 
.03244 
•03251 


•03329 
•03337 
.03345 
.03353 
03360 


•03676 
•03684 
.03692 
•03699 
•03707 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•02434 
•02440 
•02447 
•02453 
•02459 


.02494 
.02501 
.02508 
.02515 
.02521 


02831 
•02838 
•02845 

02852 
.02859 


.02914 
.02921 
.02928 
.02936 
.02943 


.03258 
•03266 
•03273 
.03281 
.03288 


03368 
•03376 
•03384 
•03392 
.03400 


03715 
•03723 
•03731 
•03739 
•03747 
•03754 
03762 
03770 
03778 
•03786 


.03858 
•03867 
.03875 
.03884 
•03892 
.03901 
.03909 
.03918 
.03927 
.03935 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


02466 
•024 72 
•02479 
•02485 
•02492 


.02528 
.02535 
.02542 
•02548 
•02555 


.02866 
.02873 
.02880 
.02887 
.02894 


.02950 
.02958 
.02965 
.02972 
.02980 


.03295 
•03303 
•03310 
.03318 
.03325 


•03408 
•03416 
.03424 
.03432 
.03439 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


02498 
•02504 
•02511 
•02517 

02524 


-02562 
.02569 
.02576 
•02582 
•02589 


•02900 
02907 
•02914 
•02921 
•02928 


.02987 
.02994 
•03002 
.03009 
.03017 


03333 
.03340 
•03347 
•03355 
•03362 


•03447 
•03455 
.03463 
•03471 
•03479 


.03794 
.03802 
•03810 
•03818 
•03826 


.03944 
.03952 
.03961 
.03969 
03978 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•02530 
•02537 
.02543 
•02550 
.02556 


•02596 
•02603 
•02610 
•02617 
•02624 


•02935 
•02942 
02949 
•02956 
•02963 


.03024 
.03032 
.03039 
03046 
.03054 


•03370 
•03377 
•03385 
.03392 
.03400 


•03487 
-03495 
•03503 
•03512 
•03520 


•03834 
.03842 
•03850 
•03858 
.03866 


.03987 
.03995 
.04004 
.04013 
.04021 


55 
56 
57 
58 
59 


60 


.02563 i 


•02630 


•02970 


.03061 


03407 


.03528 


•03874 


.04030 


60 



709 



TABLE X.- 
1€ 


-NATURAL VERSED SINES AND EXTERNAL 
17° 18° 19° 


SECANTS 


' 


Ver:^. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec 


f 

O 

1 

21! 
3 
4. - 




1 
2 
3 
4 


03874 
03882 
03890 
03898 
03906 




04030 
04039 
04047 
04056 
04065 


•04370 
•04378 
•04387 
•04395 
•04404 




04569 
04578 
04588 
04597 
04606 


.04894 
.04903 
.04912 
.04921 
•04930 




05146 
05156 
05166 
05176 
05186 


•05448 
•05458 
•05467 
.05477 
•05486 




05762 
05773 
05783 
05794 
05805 


5 
6 
7 
8 
9 


03914 
03922 
03930 
03938 
03946 




04073 
04082 
04091 
04100 
04108 


.04412 
•04421 
•04429 
•04438 
•04446 




04616 
04625 
04635 
04644 
04653 


.04939 
.04948 
.04957 
•04967 
•04976 




05196 
05206 
05216 
05226 
05236 


.05496 
.05505 
.05515 
.05524 
•05534 




05815 
05826 
05836 
05847 
05858 


5: 
6^ 
7" 
8> 

^ i 


10 

11 

12 
13 
14 


03954 1 . 
03963 ! 
03971 ! . 
03979 1 . 
03987 


04117 
04126 
04135 
04144 
04152 


•04455 
•04464 
•04472 
•04481 
•04489 

04498 
•04507 

04515 
.04524 

04533 




04663 
04672 
04682 
04691 
04700 


.04985 
.04904 
.05003 
.05012 
•05021 




05246 
05256 
05266 
05276 
05286 


.05543 
.05553 
•05562 
•05572 
•05582 




05869 
05879 
05890 
05901 
05911 


10 ^ 

11 

12 
13 
14 


15 
16 
17 
18 
19 


03995 
04003 
04011 
04019 
04028 




04161 
04170 
04179 
04188 
04197 




04710 
04719 
04729 
04738 
04748 


•05030 
•05039 
•05048 
•05057 
•05067 




05297 
05307 
05317 
05327 
05337 


•05591 
•05601 

05610 
•05620 

05630 




05922 
05933 
05944 
05955 
05965 


15 
16 
17 
18 

19 


20 

21 
22 
23 
24 


04036 
04044 
04052 
04060 
04069 




04206 
04214 
04223 
04232 
04241 


04541 
04550 
•04559 
•04567 
.04576 




04757 
04767 
04776 
04786 
04795 


•05076 
•05085 
•05094 
05103 
•05112 




05347 
05357 
05367 
05378 
05388 


•05639 
•05649 
•05658 
.05668 
•05678 




05976 
05987 
05998 
06009 
06020 


20 

21 
22 
23 
24 

25 
26 
27 
28 
29 


25 
26 
27 
28 
29 


04077 
04085 
04093 
04102 
04110 




04250 
04259 
04268 
04277 
04286 


•04585 
04593 

.04602 
04611 

.04620 




04805 
04815 
04824 
04834 
04843 


.05122 
.05131 
•05140 
•05149 
•05158 




05398 
05408 
05418 
05429 
05439 


•05687 
.05697 
05707 
•05716 
•05726 
•05736 
•05746 
•05755 
.05765 
•05775 




06030 
06041 
06052 
06063 
06074 


30 

31 
32 
33 
34 


04118 
04126 
04135 
04143 
•04151 




04295 
04304 
04313 
04322 
04331 


04628 
04637 
04646 
04655 
•04663 




04853 
04863 
04872 
04882 
04891 


.05168 
.05177 
.05186 
•05195 
.05205 




05449 
05460 
05470 
•05480 
•05490 




06085 
06096 
06107 
06118 
06129 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


04159 
.04168 
•04176 
•04184 

04193 




04340 
04349 
04358 
04367 
04376 


.04672 i 
•04681 ; 
•04690 1 
•04699 
.04707 


04901 
04911 
04920 
04930 
04940 


.05214 
.05223 
•05232 
•05242 
•05251 




•05501 
05511 
05521 
•05532 
.05542 


•05785 
•05794 
•05804 
.05814 
.05824 




06140 
06151 
0616.. 
06173 
C6184 


35 
36 
37 
38 

39 


40 

41 
42 
43 
44 


•04201 
•04209 
04218 
•04226 
•04234 




04385 
04394 
04403 
04413 
04422 


04716 
•04725 
•04734 
•04743 
.04752 




04950 
04959 
04969 
04979 
.04989 


•05260 
•05270 
.05279 
•05288 
•05298 




•05552 
05563 

■05573 
05584 
05594 


05833 
•05843 
•05853 

05863 
•05873 




06195 
05206 
•06217 
06228 
06239 


40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 


45 
46 
47 
43 
49 


•04243 
•04251 
04260 
■04268 
•04276 




04431 
04440 
04449 
04458 
04468 


•04760 
04769 
•04778 
•04787 
04796 




04998 
05008 
05018 
05028 
•05038 


.05307 
.05316 
.05326 
.05335 
.05344 




05604 
05615 
.05625 
.05636 
.05646 


•05882 
•05892 
•05902 
.05912 
•05922 




06250 
06261 
06272 
06283 
06295 


50 

51 
52 
53 
54 


04285 
•04293 
•04302 
•04310 
•04319 




04477 
04486 
04495 
04504 
04514 


04805 
•04814 
•04823 
•04832 

04841 

04850 
04858 
•04867 
•04876 
04885 




.05047 
•05057 
05067 
•05077 
•05087 


.05354 
.05363 
.05373 
.05382 
05391 




05657 
05667 
05678 
05688 
05699 


•05932 
•05942 
•05951 
.05961 
•05971 




06306 
06317 
06328 
06339 
06350 


55 
56 
57 
58 
59 


•04327 
•04336 
•04344 
04353 
•04361 




•04523 
•04532 
•04541 
•04551 
•04560 




.05097 
•05107 
•05116 
•05126 
•05136 


.05401 
.05410 
.05420 
.05429 
•05439 




•05709 
05720 
05730 
05741 
05751 


05981 
•05991 
•06001 
•06011 
•06021 




06362 
06373 
06384 
06395 
06407 


60 


•04370 




04569 


•04894 




•05146 


•05448 




.05762 


.06031 




06418 


60 



710 



TAB1.E X.— 
30° 


NATURAL VERSED SINES AND EXTERNAL SECANTS. 
31° 33° 33° 


' 


Vers. 


Exc sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


' 




1 
2 
3 
4 

5 
6 
7 
8 
9 


.G6031 
•06041 
•06051 
•06061 
•06071 




06418 
06429 
06440 
06452 
06463 


-06642 
•06652 
•06663 
•06673 
•06684 




07115 
07126 
07138 
07150 
07162 


.07282 
•07293 
.07303 
.07314 
.07325 




07853 
07866 
07879 
07892 
07904 


■07950 
•07961 
•07972 
•07984 
■07995 




08636 
08649 
08663 
08676 
08690 




1 
2 
3 
4 


06081 
•06091 
•06101 
•06111 

06121 


06474 
06486 
06497 
06508 
06520 


•06694 
•06705 
•06715 
•06726 
•06736 


- 


07174 
07186 
07199 
07211 
07223 


•07336 
•07347 
-07358 
•07369 
07380 




07917 
07930 
07943 
07955 
07968 


08006 
■08018 
•08029 
•08041 
■08052 




08703 
08717 
08730 
08744 
08757 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•06131 
•06141 
•06151 
•06161 
-06171 




06531 
06542 
06554 
06565 
06577 


06747 
•06757 
■06768 
•06778 
-06789 


07235 
07247 
07259 
07271 
07283 


•07391 
•07402 
■07413 
•07424 
.07435 




07981 
07994 
08006 
08019 
08032 


•08064 
•08075 
•08086 
•08098 
-08109 
•08121 
.08132 
.08144 
•08155 
08167 


- 


08771 
08784 
08798 
08811 
08825 
08839 
08852 
08866 
08880 
08893 


10 

11 

12 
13 
14 


15 
16 
17 
18 
19 
30 
21 
22 
23 
24 

25 
26 
27 
28 
29 


■06181 
•06191 
•06201 
•06211 
•06221 




06588 
06600 
06611 
06622 
06634 


-06799 
•06810 
•06820 
•06831 
-06841 




07295 
07307 
07320 
07332 
07344 


.07446 
•07457 
.07468 
.07479 
.07490 




08045 
08058 
08071 
08084 
08097 


15 
15 
17 
18 
19 


•06231 
•06241 
•06252 
•06262 
•06272 




06645 
06657 
06668 
06680 
06691 


-06852 
•06863 
.06873 
•06884 
•06894 




07356 
07368 
07380 
07393 
07405 


.07501 
.07512 
•07523 
•07534 
•07545 




08109 
08122 
08135 
08148 
08161 


.08178 
.08190 
.08201 
•08213 
•08225 




08907 
08921 
08934 
08948 
08962 


30 

21 
22 
23 
24 


06282 
•06292 
•06302 
•06312 
•06323 




06703 
06715 
06726 
06738 
06749 


•06905 
•06916 
•06926 
.06937 
.06948 




07417 
07429 
07442 
07454 
07466 


•07556 
.07568 
.07579 
.07590 
•07601 




08174 
08087 
08200 
08213 
08226 


08236 
08248 
•08259 
.08271 
08282 




08975 
08989 
09003 
09017 
.09030 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 

35 
36 
37 
38 
39 


•08333 
•06343 
•06353 
•06363 
06374 




06761 
06773 
06784 
06796 
06807 


.06958 
.06969 
.06980 
.06990 
.07001 




07479 
07491 
07503 
07516 
07528 


•07612 
07623 
•07634 
•07645 
•07657 




08239 
08252 
.08265 
08278 
08291 


•08294 
•08306 
•08317 
•08329 
08340 




09044 
.09058 
.09072 
.09086 
.09099 


30 

31 

32 
33 
34 


06384 
06394 
•06404 
•06415 
06425 




06819 
06831 
06843 
06854 
06866 


.07012 
•07022 
.07033 
-07044 
•07055 




07540 
07553 
07565 
07578 
07590 


•07668 
•07679 
•07690 
07701 
•07713 




08305 
.08318 
.08331 
08344 
08357 


•08352 
•08364 
•08375 
•08387 
•08399 




.09113 
09127 
09141 
09155 
09169 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•06435 
•06445 
06456 
•06466 
-06476 




06878 
06889 
06901 
06913 
06925 


•07065 
•07076 
•07087 
07098 
•07108 




07J02 
07615 
07627 
07b40 
07652 


.07724 
.07735 
.07746 
.07757 
.07769 




08370 
08383 
08397 
08410 
08423 


.08410 
•08422 
08434 
.08445 
•08457 




09183 
09197 
09211 
09224 
09238 


40 

41 
42 
48 
44 


45 
46 
47 
48 
49 


06486 
•06497 
•06507 
•06517 
•06528 




06936 
06948 
06960 
06972 
06984 


•07119 
•07130 
•07141 
•07151 
•07162 




07665 
07677 
07690 
07702 
07715 


•07780 

•07791 

•07802 

07814 

J)7825 

.07836 

07848 

07859 

.07870 

.07881 




08436 
08449 
08463 
08476 
08489 

08503 
08516 
08529 
08542 
08556 


08469 
.08481 
.08492 
•08504 
.08516 




09252 
09266 
09280 
09294 
09308 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•06538 
•06548 
•06559 
06569 
•06580 




06995 
07007 
07019 
07031 
07043 


•07173 
.07184 
.07195 
.07206 
.07216 




07727 
07740 
07752 
07765 
07778 


.08528 
.08539 
.08551 
.08563 
.08575 




09323 
09337 
09351 
09365 
09379 


50 

51 

52 
53 
54 


55 
56 
57 
58 
59 


•06590 
•06600 
06611 
•06621 
•06632 




07055 
07067 
07079 
07091 
07103 


•07227 
•07238 
•07249 
.07260 
•07271 




07790 
07803 
07816 
07828 
07841 


.07893 
.07904 
•07915 
•07927 
07938 




08569 
08582 
08596 
08069 
08623 


•08586 
08598 
08610 

•08622 
08634 




09393 
09407 
09421 
09435 
09449 


55 
56 
57 
58 
59 


60 


06642 




07115 


•07282 




07853 


•07950 


_ 


08636 


•08645 j .09464 


60 



711 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
34° 25° 36° 27° 



' 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. SRC. 


' 




1 

2 
3 
4 


08645 
.08657 
.08669 
.08681 
.08693 


.09464 
.09478 
.09492 
.09506 
•09520 
•09535 
.09549 
•09563 
.09577 
.09592 


.09369 
•09382 
.09394 
.09406 
•09418 


.10338 
•10353 
.10368 
•10383 
.10398 


10121 
.10133 
.10146 
.10159 
■10172 


•11260 
.11276 
•11292 
•11308 
.11323 


•10899 
•10913 
•10926 
.10939 
■10952 


.12233 
.1224b 
•12266 
•12283 
-12299 


O 

1 
2 
3 
4 


5 
6 
7 
8 
9 


.08705 
.08717 
.08728 
.08740 
.08752 


.09431 
•09443 
•09455 
09468 
•09480 


•10413 
•10428 
• 10443 
•10458 
.10473 


•10184 
•10197 
.10210 
•10223 
•10236 


•11339 
11355 
•11371 
•11387 
■11403 


■10965 
.10979 
■10992 
•11005 
■11019 


•12316 
•12333 
•12349 
•12366 
•12383 


5 

6 
7 
8 
9 


10 

11 
12 
13 
14 


.08764 
.08776 
.08788 
.088C0 
.08812 


.09606 
.09620 
.09635 
.09649 
.09663 


.09493 
.09505 
•09517 
.09530 
.09542 


.10488 
•10503 
10518 
.10533 
.10549 


•10248 
•10261 
•10274 
•10287 
•10300 


•11419 
•11435 
-11451 
•11467 
.11483 


.11032 
.11045 
•11058 
11072 
.11085 


• 12400 
•12416 
•12433 
•12450 
•12467 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.08824 
.08836 
.08848 
.08860 
08872 


.09678 
.09u92 
.09707 
■09721 
^09735 


09554 
•09567 
.09579 
.09592 
.09604 


.10564 
.10579 
•10594 
.10609 
•10625 


•10313 
•10326 
•10338 
.10351 
.10364 


•11499 
11515 
•11531 
•11547 
•11563 


■11098 
■11112 
■11125 
■11138 
■11152 


• 12484 
•12501 
•12518 
-12534 
-12551 


15 
16 
17 
18 

19 


20 

21 
22 
23 
24 


.08884 
.08896 
.08908 
.08920 
.08932 


.09750 
•09764 
•09779 
•09793 
.09808 


•09617 
.09629 
.09642 
.09654 
.09666 


•10640 
.10655 
.10670 
•10686 
•10701 


•10377 
•10390 
•10403 
.10416 
10429 


•11579 
•11595 
■11611 
•11627 
11643 


■11165 
■11178 
.11192 
•11205 
-11218 


-12568 
•12585 
•12602 
-12619 
12636 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.08944 
.08956 
.08968 
•08980 
08992 


.09822 
•09837 
•09851 
.09866 
•09880 


•09679 
•09691 
•09704 
•09716 
•09729 


.10716 
.10731 
•10747 
•10762 
.10777 


• 10442 
•10455 
.10468 
.10481 
•10494 


■11659 
•11675 
.11691 
.11708 
.11724 


•11232 
•11245 
-11259 
-11272 
.11285 


•12653 
•12670 
•12687 
•12704 
12721 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.09004 
.09016 
.09028 
.09040 
.09052 


•09895 
•09909 
.09924 
09939 
•09953 


.09741 
•09754 
.09767 
.09779 
.09792 


•10793 

10808 

10824 

•10839 

.10854 


.10507 
•10520 
•10533 
-10546 
.10559 


.11740 
•11756 
.11772 
.11789 
11805 


•11299 
-11312 
-11326 
-11339 
.11353 


•12738 
-12755 
■12772 
-12789 
■12807 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.09064 
.09076 
.09089 
.09101 
.09113 


.09968 
•09982 
•09997 
•10012 
•10026 


•09804 
09817 
09829 
•09842 
•09854 


.10870 
.10885 
.10901 
•10916 
•10932 


•10572 
•10585 
•10598 
■10611 
.10624 


.11821 

11838 

■11854 

■11870 

11886 


.11366 
.11380 
.11393 
.11407 
■11420 


-12824 
•12841 
-12858 
•1^875 
-12892 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•09125 
.09137 
•09149 
•09161 
•09174 


.10041 
.10055 
•10071 
.10085 
.10100 


•09867 
•09880 
•09892 
•09905 
•09918 


•10947 
.10963 
•10978 
•10994 
ai009 


■10637 
10650 
■10663 
•10676 
•10689 
•10702 
•10715 
•10728 
•10741 
•10755 


■11903 
.11919 
.11936 
■11952 
.11968 


■11434 
■11447 
-11461 
-11474 
-11488 


12910 
•12927 
•12944 
•12961 
-12979 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•09186 
09198 
•09210 
•09222 
•09234 


.10115 
.10130 
.10144 
.10159 
•10174 


•09930 
•09943 
•09955 
•09968 
09981 


•11025 
•11041 
•11056 
•11072 
.11087 


.11985 
-12001 
-12018 
-12034 
■12051 


-11501 
-11515 
■11528 
•11542 
•11555 


-12996 
•13013 
-13031 
•13048 
13065 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.09247 
09259 
•09271 
•09283 
•09296 


.10189 
-10204 
.10218 
.10233 
.10248 


•09993 
•10006 
■10019 
•10032 
• 10044 


•11103 
.11119 
.11134 
.11150 
•11166 


■10768 
•10781 

10794 
■10807 

10820 


•12067 
■ 12084 
-12100 
•12117 
.12133 


-11569 
-11583 
■11596 
11610 
-11623 


-13083 
-13100 
-13117 
-13135 
13152 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•09308 
.09320 
.09332 
■09345 
•09357 


.10263 
.10278 
.10293 
.10308 
.10323 

.10338 


•10057 
•10070 
•10082 
•10095 
10108 
10121 


.11181 
.11197 
.11213 
.11229 
•11244 


•10833 
•10847 
•10860 
•10873 
.10886 


.12150 
.12166 
•12183 
.12199 
•12216 


•11637 
•11651 
.11664 
.11678 
-11692 


-13170 
•13187 
•13205 
-13222 
-13240 


55 
56 
57 
58 

59 


60 


•09369 


.11260 


•10899 


.12233 


-11705 


■13257 


60 










l-ri 













712 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
38° 39° 30° 31° 


' 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


' 




1 
2 
3 
4 


.11705 
.11719 
.11733 
.11746 
.11760 


.13257 
.13275 
.13292 
.13310 
.13327 


.12538 
,.12552 
•12566 
.12580 
.12595 




14335 
14354 
14372 
14391 
14409 


•13397 
•13412 
.13427 
•13441 
•13456 


.15470 
.15489 
.15509 
.15528 
.15548 


■ 14283 
.14298 
.14313 
•14328 

■ 14343 


•16663 
•16684 
•16704 
.16725 
•16745 




1 
2 
3 
4 


5 
6 
7 
8 
9 

10 

11 

12 

13 

14 

15 
16 
17 
18 
19 


.11774 
.11787 
.11801 
.11815 
.11828 


.13345 
.13362 
.13380 
.13398 
.13415 


.12609 
.12623 
.12637 
.12651 
•12665 




14428 
14446 
14465 
14483 
14502 


•13470 
.13485 
.13499 
.13514 
.13529 


.15567 
•15587 
•15606 
■15626 
.15645 


■14358 

■ 14373 
■14388 

■ 14403 
•14418 


•16766 
•16786 
•16806 
•16827 
•16848 


5 

6 
7 
8 
9 


.11842 
.11856 
.11870 
.11883 
.11897 


. 13433 
.13451 
.13468 
.13486 
•13504 


.12679 
.12694 
.12708 
.12722 
•12736 




14521 
14539 
14558 
14576 
14595 


.13543 
.13558 
.13573 
.13587 
•13602 


15665 
•15684 
.15704 
.15724 
■15743 


■ 14433 

■ 14449 

■ 14464 

■ 14479 

■ 14494 


.16868 
.16889 
•16909 
•16930 
■16950 


10 

11 
12 
13 
14 


.11911 
.11925 
•11938 
.11952 
•11966 


■13521 
•13539 
.13557 
•13575 
.13593 
•13610 
.13628 
.13646 
.13664 
.13682 


.12750 
.12765 
.12779 
.12793 
•12807 
.12822 
.12836 
.12850 
.12864 
.12879 




.14614 
.14632 
14651 
14670 
14689 


•13616 
•13631 
•13646 
13660 
•13675 


.15763 
.15782 
.15802 
.15822 
.15841. 


■14509 
■14524 
■14539 
■14554 
•14569 


■16971 
■16992 
■17012 
•17033 
.17054 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 

25 
26 
27 
28 
29 


.11980 
.11994 
.12007 
.12021 
.12035 




14707 
•14726 
14745 
14764 
14782 


•13690 
■13705 
■13719 
■13734 
■13749 


.15861 
.15881 
.15901 
.15920 
•15940 


■14584 
■14599 
.14615 
.14630 
• 14645 


■17075 
.17095 
■17116 
■17137 
•17158 


20 

21 
22 
23 
24 


.12049 
.12063 
•12077 
•12091 
12104 


-13700 
.13718 
.13735 
.13753 
.13771 


.12893 
.12907 
•12921 
.12936 
.12950 




14801 
14820 
14839 
14858 
14877 


•13763 
•13778 
•13793 
•13808 
.13822 


.15960 
•15980 
•16000 
.16019 
•16039 


•14660 
.14675 
.14690 
.14706 
.14721 


■17178 
•17199 
.17220 
•17241 
■17262 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.12118 

•12132 

-12146 

12160 

12174 


.13789 
.13807 
.13825 
.13843 
.13861 


.12964 
•12979 
•12993 
.13007 
.13022 




14896 
14914 
14933 
14952 
14971 


.13837 
.13852 
.13867 
.13881 
■13896 


.16059 
.16079 
.16099 
.16119 
.16139 


.14736 
.14751 
.14766 
•14782 
.14797 


■17283 
■17304 
■17325 
■17346 
•17367 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.12188 
•12202 
•12216 
•12230 
•12244 


.13879 
.13897 
.13916 
.13934 
13952 


.13036 
.13051 
.13065 
.13079 
•13094 




14990 
15009 
15028 
15047 
15066 


•13911 
■13926 
■13941 
■13955 
.:i8970 


.16159 
•16179 
.16199 
■16219 
■16239 


.14812 i .17388 
.14827 1 .17409 
.14843 .17430 
.14858 ! .17451 
■14873 .17472 


35 
36 
37 
38 
-19 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 


.12257 
.12271 
.12285 
.12299 
.12313 


.13970 
.13988 
•14006 
.14024 
. 14042 


•13108 
•13122 
.13137 
.13151 
•13166 




15085 
1510b 
15124 
15143 
15162 


13985 

• 14000 
■14015 

• 14030 
•14044 


.16259 
■16279 
•16299 
•16319 
•16339 


.14888 
.14904 
.14&19 
.14934 
■14949 


•17493 
•17514 
■17535 
.17556 
.17577 


40 

41 
42 
43 
44 


.12327 
.12341 
.12355 
.12369 
12383 


.14061 
. 14079 
.14097 
•14115 
•14134 


.13180 
13195 
.13209 
.13223 
•13238 




15181 
15200 
15219 
15239 
15258 


14059 

14074 

•14089 

■14104 

•14119 


16359 
•16380 
•16400 
•16420 
•16440 


•14965 
•14980 
•14995 
•15011 
•15026 


.17598 
.17620 
.17641 
.17662 
• 1768-3 


45 
46 
47 
48 
49 


• 1239*7 
.12411 
.12425 
. 12439 
•12454 


.14152 
.14170 
.14188 
.14207 
.14225 


•13252 
.13267 
.13281 
.13296 
.13310 




15277 
15296 
15315 
15335 
15354 


■14134 
.14149 
.14164 
•14179 
■14194 


• 16460 
16481 
.16501 
.16521 
•16541 


15041 
.15057 
.15072 
.15087 
■15103 


•17704 
•17726 
•17747 
.17768 
•17790 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 
60 


.12468 
.12482 
.12496 
.12510 
.12524 


.14243 
14262 
•14280 
.14299 
•14317 


.13325 
•13339 
•13354 
•13368 
.13383 




15373 
15393 
15412 
15431 
15451 


.14208 
.14223 
•14238 
•14253 
•14268 


.16562 
.16582 
■16602 
■16623 
16643 


15118 
.15134 
■15149 
.15164 

15180 


.17811 
•17832 
•17854 
•17875 
.17896 


55 
56 
57 
58 
59 


.12538 


•14335 


.13397 1 


15470 


.14283 


.16663 


■15195 


.17918 


60 



713 



TABLE X.— 
32' 


NATURAL VERSED SINES AND EXTERNAL 
33° 34° 35 


SECANTS 


' 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


. ! 




1 
2 
3 
4 


.15195 
.15211 
.15226 
.15241 
.15257 




17918 
17939 
17961 
17982 
18004 


.16133 
.16149 
•16165 
.16181 
.16196 




19236 
19259 
19281 
19304 
•19327 


.17096 
.17113 
.17129 
.17145 
•17161 




•20622 
20645 
•20669 
•20693 
•20717 


.18085 
.18101 
.18118 
•18135 
•18152 




22077 
22102 
22127 
.22152 
•22177 




1 

2 
3 
4 


5 
6 
7 
8 

9 


.15272 
.15288 
.15303 
.15319 
.15334 




18025 
18047 
18068 
18090 
18111 


.16212 
.16228 
.16244 
.16260 
.16276 




19349 

19372 

•19394 

•19417 

• 19440 


•17178 
•17194 
•17210 
.17227 
•17243 




.20740 
.20764 
20788 
.20812 
•20836 


.18168 
18185 
.18202 
.18218 
.18235 




22202 
•22227 
.22252 

22277 
.22302 


3 
6^ 
7' 

8 

9 


10 

11 
12 
13 
14 


.15350 
.15365 
.15381 
.15396 
•15412 




18133 
18155 
18176 
18198 
18220 


•16292 
•16308 
•16324 
•16340 
•16355 




19463 
19485 
19508 
19531 
19554 


•17259 
•17276 
■17292 
•17308 
•17325 




•20859 

•20883 

20907 

20931 

20955 


.18252 
.18269 
.18286 
.18302 
.18319 




22327 
•22352 
•22377 
•22402 
.22428 


10' 

11 
12 
13; 

14 


15 
16 
17 
18 
19 


•15427 
•15443 
.15458 
•15474 
15489 
•15505 
•15520 
•15536 
•15552 
•15567 


- 


18241 
18263 
18285 
18307 
18328 
18350 
18372 
18394 
18416 
18437 


•16371 
16387 
16403 

•16419 
16^35 

•16451 

•16467 

16483 

16499 

16515 




19576 
19599 
19622 
19645 
19668 


•17341 

•17357 

•17374 

17390 

17407 




20979 
.21003 
■21027 
.21051 

21075 


•18336 
•18353 
.18369 
•18386 
•18403 




.22453 
22478 
22503 
22528 
22554 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 




19691 
19713 
19736 
19759 
19782 


•17423 
•17439 
•17456 
•17472 
•17489 




21099 
21123 
21147 
21171 
21195 


•18420 
18437 
.18454 
.18470 
.18487 




22579 
22604 
22629 
22655 
22680 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•15583 

.15598 

15614 

15630 

15645 




18459 
18481 
18503 
18525 
18547 


16531 
16547 
16563 
16579 
16595 




19805 
19828 
19851 
19874 
19897 


•17505 
•17522 
•17538 
•17554 
.17571 




21220 
21244 
21268 
21292 
21316 


.18504 
•18521 
18538 
.18555 
.18572 




22706 
22731 
22756 
22782 
22807 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•15661 
•15676 
•15692 
•15708 
•15723 




18569 
18591 
18613 
18635 
18657 


•16611 
16627 
16644 
16660 
16676 




19920 
19944 
19967 
19990 
20013 


•17587 
•17604 

17620 
•17637 

17653 




21341 
21365 
21389 
21414 
21438 


•18588 
•18605 
■18622 
■18639 
■18656 




22833 
22858 
22884 
22909 
22935 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


15739 
15755 
•15770 
•15786 
•15802 




18679 
18701 
18723 
18745 
18767 


16692 
16708 

■16724 
16740 

.16756 




20036 
20059 
20083 
20106 
20129 


17670 
17686 
17703 
17719 
17736 




21462 
21487 
21511 
21535 
21560 


■18673 

18690 

■18707 

•18724 

18741 




22960 
22986 
23012 
23037 
23063 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•15818 
15833 
•15849 
•15865 
.15880 




18790 
18812 
18834 
18856 
18878 


•16772 
•16788 
•16805 
•16821 
•16837 




20152 
20176 
20199 
20222 
20246 


•17752 

17769 

17786 

■17802 

•17819 




21584 
21609 
21633 
21658 
21682 


■18758 

18775 

18792 

■18809 

.18826 




23089 
23114 
23140 
23166 
23192 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•15896 
•15912 
•15928 
•15943 
15959 




18901 
18923 
18945 
18967 
18990 


16853 
•16869 

16885 
•16902 
•16918 




20269 
20292 
20316 
20339 
20363 


■17835 
•17852 
17868 
•17885 
•17902 




21707 
21731 
21756 
21781 
21805 


■18843 

18860 

18877 

■18894 

■18911 




23217 
23243 
23269 
23295 
23321 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•15975 
15991 
•16006 
•16022 
•18038 




19012 
19034 
19057 
19079 
19102 


16934 
•16950 
16966 
16983 
16999 




20386 
20410 
20433 
20457 
20480 


■17918 
•17935 
•17952 
17968 
17985 




21830 
21855 
21879 
21904 
21929 


18928 
■18945 
18962 
18979 
18996 




23347 
23373 
23399 
23424 
23450 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•16054 
•16070 
.16085 
•16101 
•16117 




19124 
19146 
19169 
19191 
19214 


•17015 

17031 

17047 

.17064 

.17080 




20504 
20527 
20551 
20575 
20598 


■18001 
18018 
•18035 
•18051 
•18068 




21953 
21978 
22003 
22028 
22053 


.19013 
.19030 
.19047 
•19064 
•19081 




23476 
23502 
23529 
23555 
23581 


55 
56 
57 
58 
59 


60 


.16133 




19236 


.17096 




20622 


•18085 




22077 


.19098 




23607 


60 



714 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS 
36° 37° 38° 39° 



t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 

4 


•19098 
.19115 
.19133 
.19150 
.19167 


•23607 
•23633 
23659 
•23685 
•23711 


.20136 
.20154 
.20171 
.20189 
.20207 




25214 
25241 
25269 
25296 
25324 


.21199 
•21217 
•21235 
.21253 
.21271 


- 


26902 
26931 
26960 
26988 
27017 
27046 
27075 
27104 
27133 
27162 


22285 
22304 
22322 
.22340 
.22359 




28676 
28706 
28737 
28767 
28797 




1 
2 
3 
4 


5 
6 
7 
8 

9 


.19184 
.19201 
.19218 
.19235 
.19252 


.23738 
.23764 
.23790 
.23816 
.23843 


.20224 
.20242 
.20259 
.20277 
.20294 


- 


25351 
25379 
25406 
25434 
25462 


.21289 
.21307 
.21324 
.21342 
.21360 


•22377 
•22395 
•22414 
•22432 
•22450 




28828 
28858 
28889 
28919 
28950 


5 
6 
7 
8 
9 


10 

11 
12 
13 

14 


.19270 
.19287 
19304 
.19321 
.19338 


.23869 
.23895 
.23922 
.23948 
.23975 


.20312 
.20329 
.20347 
.20365 
.20382 


25489 
25517 
25545 
25572 
25600 


•21378 
•21396 
.21414 
•21432 
•21450 




27191 
27221 
27250 
27279 
27308 


•22469 
•22487 
•22506 
•22524 
•22542 




28980 
29011 
29042 
29072 
29103 


10 

11 

12 
13 
14 


15 
16 
17 
18 
19 


19356 
19373 
19390 
19407 
19424 


.24001 
.24028 
. 24054 
.24081 
•24107 


.20400 
.20417 
.20435 
.20453 
.20470 




25628 
25656 
25683 
25711 
25739 


•21468 
•21486 
•21504 
.21522 
•21540 




27337 
27366 
27396 
27425 
27454 


•22561 
•22579 
22598 
•22616 
•22634 




29133 
29164 
29195 
29226 
29256 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


.19442 
.19459 
.19476 
.19493 
.19511 


.24134 
•24160 
.24187 
.24213 
. 24240 


.20488 
•20506 
•20523 
•20541 
•20559 




25767 
25795 
25823 
25851 
25879 


•21558 
•21576 
•21595 
.21613 
•21631 




27483 
27513 
27542 
27572 
27601 


•22653 
•22671 
•22690 
•22708 
•22727 




29287 
29318 
29349 
29380 
29411 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.19528 
.19545 
•19562 
.19580 
•19597 


.24267 
.24293 
•24320 
•24347 
•24373 


•20576 
•20594 
•20612 
•20629 
•20647 




25907 
25935 
25963 
25991 
26019 


•21649 
•21667 
•21685 
•21703 
•21721 




27630 
27660 
27689 
27719 
27748 


•22745 
•22764 
•22782 
•22801 
•22819 




29442 
29473 
29504 
29535 
29566 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•19614 
•19632 
•19649 
•19666 
•19684 


• 24400 
.24427 
•24454 
•24481 
•24508 


.20665 
•20682 
•20700 
•20718 
•20736 




26047 
26075 
26104 
26132 
26160 


•21739 
•21757 
•21775 
•21794 
•21812 




27778 
27807 
27837 
27867 
27896 


•22838 
•22856 
22875 
•22893 
•22912 




29597 
29628 
29659 
29690 
29721 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•19701 
•19718 
.19736 
•19753 
•19770 


•24534 
•24561 
•24588 
•24615 
. 24642 


.20753 
•20771 
•20789 
•20807 
•20824 




26188 
26216 
26245 
26273 
26301 


•21830 
•21848 
•21866 
-21884 
•21902 




27926 
27956 
27985 
28015 
28045 


•22930 
•22949 
•22967 
•22986 
• 23004 




29752 
29784 
29815 
29846 
29877 


35 
36 
37 
38 
-39 


40 

41 
42 
43 
44 


•19788 
.19805 
•19822 
•19840 
•19857 


■24669 
24696 
-24723 
•24750 
•24777 


•20842 
•20860 
•20878 
•20895 
•20913 




26330 
26358 
26387 
26415 
26443 


•21921 
21939 
•21957 
•21975 
•21993 




28075 
28105 
.28134 
.28164 
•28194 


•23023 
23041 
•23060 
.23079 
.23097 




29909 
29940 
29971 
30003 
30034 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


•19875 
•19892 
•19909 
•19927 
•19944 


• 24804 
•24832 
24859 
•24886 
•24913 


•20931 
•20949 
•20967 
•20985 
•21002 




26472 
26500 
26529 
26557 
26586 


•22012 
-22030 
•22048 
•22066 
•22084 




-28224 
-28254 
•28284 
•28314 
•28344 


23116 
•23134 
•23153 
•23172 
•23190 




30066 
30097 
30129 
.30160 
.30192 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•19962 
•19979 
•19997 
•20014 
.20032 


. 24940 
.24967 
•24995 
•25022 
. 25049 


•21020 
21038 
•21056 
•21074 
•21092 




26615 
26643 
26672 
.26701 
•26729 


•22103 
•22121 
•22139 
•22157 
•22176 




•28374 
•28404 
•28434 
.28464 
28495 


•23209 
•23228 
•23246 
•23265 
23283 




.30223 
.30255 
.30287 
.30318 
.30350 


50 

51 
52 
53 
54 


55 
56 
57 
58 
5^ 


. 20049 
•20066 
.20084 
.20101 
.20119 


•25077 
.25104 
.25131 
•25159 
25186 


•21109 
•21127 
•21145 
•21163 
•21181 




.26758 
26787 
26815 
•26844 
•26873 


•22194 
•22212 
•22231 
• 22249 
•22267 




•28525 
•28555 
•28585 
•28615 
•28646 


•23302 
•23321 
•23339 
•23358 
•23377 




.30382 
.30413 
•30445 
•30477 
30509 


55 
56 
57 
56 
59 


60 


.20136 


.25214 


.21199 




.26902 


.22285 




.28676 


.23396 




.30541 


60 



715 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL 
40° 41° 42° 43 


SECANT.^ ^ 

o 

- 


/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 




1 
2 
3 
4 


•23396 
.23414 
.23433 
.23452 
•23470 


.30541 
.30573 
.30605 
.30636 
.30668 

.30700 
.30732 
.30764 
.30796 
.30829 


.24529 
.24548 
.24567 
.24586 
•24605 


.32501 
.32535 
.32568 
.32602 
•32636 


.25686 
.25705 
.25724 
•25744 
.25763 




34563 
.34599 
34634 
34669 
34704 


.26865 
•26884 
.26904 
.26924 
.26944 




.36733 
.36770 
.36807 
.36844 
•36881 


1 
2 
3 
4 - 


5 
6 
7 
8 
9 


.23489 
.23508 
.23527 
.23545 
.23564 


.24625 
■24644 
.24663 
•24682 
•24701 


.32669 
.32703 
.32737 
.32770 
.32804 

.32838 
.32872 
.32905 
.32939 
.32973 

.33007 
.33041 
.33075 
.33109 
•33143 


.25783 
.25802 
.25822 
•25841 
•25861 




34740 
34775 
34811 
34846 
34882 


.26964 
.26984 
.27004 
.27024 
.27043 




.36919 
.36956 
•36993 
•37030 
•37068 


5 
6' 
7 
8 
9 - 


10 

11 
12 
13 
14 


.23583 
.23602 
.23620 
.23639 
•23658 


.30861 
.30893 
.30925 
•30957 
•30989 

.31022 
.31054 
.31086 
.31119 
.31151 


•24720 
.24739 
•24759 
•24778 
•24797 
.24816 
•24835 
.24854 
•24874 
.24893 


•25880 
•25900 
.25920 
.25939 
.25959 
.25978 
•25998 
.26017 
.26037 
.26056 




34917 
34953 
34988 
35024 
35060 


27063 
.27083 
.27103 
.27123 
•27143 




•37105 
•37143 
37180 
37218 
37255 


10 ) 

11 
12. 

13 
14 ^ 


15 
16 
17 
18 
19 


.23677 
.23696 
•23714 
•23733 
23752 




35095 
35131 
35167 
35203 
35238 


.27163 
•27183 
.27203 
•27223 
•27243 




37293 
37330 
37368 
37406 
37443 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


■23771 
.23790 
.23808 
.23827 
23846 

23865 
23884 
.23903 
23922 
23941 


.31183 
.31216 
.31248 
.31281 
•31313 


•24912 
•24931 
•24950 
24970 
.24989 


.33177 
.33211 
.33245 
.33279 
.33314 
.33348 
.33382 
.33416 
.33451 
.33485 


.26076 
.26096 
.26115 
.26135 
.26154 




35274 
35310 
35346 
35382 
35418 


.27263 
•27283 
•27303 
.27323 
•27343 


- 


37481 
37519 
37556 
37594 
37632 
37670 
37708 
37746 
37784 
37822 


20 

21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 


25 
26 
27 
28 
29 


.31346 
.31378 
.31411 
.31443 
.31476 


.25008 
.25027 
.25047 
.25066 
.25085 


.26174 
.26194 
.26213 
.26233 
.26253 




35454 
35490 
35526 
35562 
35598 


•27363 
.27383 
.27403 
.27423 
.27443 


30 

31 
32 
33 
34 


23959 
•23978 
23997 
24016 
24035 


.31509 
.31541 
.31574 
.31607 
.31640 


•25104 
.25124 
25143 
25162 
.25182 


.33519 
.33554 
.33588 
.33622 
.33657 


.26272 
.26292 
.26312 
.26331 
.26351 




35634 
35670 
35707 
35743 
35779 


.27463 
•27483 
•27503 
•27523 
•27543 




37860 
37898 
37936 
37974 
38012 


35 
36 
37 
38 
39 


24054 
•24073 
24092 
24111 
24130 


.31672 
.31705 
.31738 
.31771 
.31804 


.25201 
.25220 
.25240 
.25259 
•25278 


•33691 
•33726 
.33760 
.33795 
.33830 


•26371 
•26390 
•26410 
•26430 
.26449 




35815 
35852 
35888 

35924 
35961 


.27563 
•27583 
•27603 
.27623 
.27643 




38051 
38089 
38127 
38165 
38204 


35 
36 
37 
38 
39 

40 

41 
42 
43 
44 


40 

41 
42 
43 
44 


24149 
.24168 
.24187 
.24206 

24225_ 

24244 
• 24262 
.24281 
.24300 

24320 
.24339 
.24358 
.24377 
.24396 
.24415 


.31837 
.31870 
.31903 
•31936 
.31969 


•25297 
•25317 
.25336 
•25356 
•25375 


•33864 
.33899 
.33934 
.33968 
.34003 


•26469 
•26489 
•26509 
•26528 
.26548 




35997 
36034 
36070 
36107 
36143 


•27663 
•27683 
•27703 
.27723 
.27743 


- 


38242 
38280 
38319 
38357 
38396 


45 
46 
47 
48 
49 


.32002 
.32035 
.32068 
.32101 
.32134 


.25394 
.25414 
.25433 
•25452 
.25472 


.34038 
•34073 
•34108 
•34142 
•34177 


.26568 
.26588 
•26607 
.26627 
•26647 




36180 
36217 
36253 
36290 
36327 


.27764 
.27784 
.27804 
.27824 
.27844 


38434 
38473 
38512 
38550 
38^9_ 

38628 
38666 
38705 
38744 
38783 


45 
46 
47 
48 
49 
50 
51 
52 
53 
54 


50 

51 
52 
53 
54 


.32168 

.32201 
.32234 
•32267 
.32301 


.25491 
.25511 
.25530 
.25549 
.25569 


•34212 
.34247 
.34282 
.34317 
.34352 


.26667 
•26686 
•26706 
•26726 
•26746 




36363 
36400 
36437 
36474 
36511 


.27864 
.27884 
.27905 
.27925 
.27945 


55 
56 
57 
58 
59 


.24434 
.24453 
.24472 
.24491 
.^4510 
•24529 


.32334 
.32368 
.32401 
.32434 
.32468 


•25588 
.25608 
.25627 
.25647 
.25666 


.34387 

.34423 
.34458 
.34493 
.34528 


•26766 
•26785 
.26805 
.26825 
•26845 




36548 
36585 
36622 
36659 
36696 


•27965 
.27985 
28005 
•28026 
.28046 




38822 
38860 
38899 
38938 
38977 


55 
56 
57 
58 
59 


60 


.32501 


.25686 


.34563 


.26865 




36733 


.28066 




39016 


60 



716 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
44° 45° 46° 47° 



/ 



1 

1 

4 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. Ex. sec. 


/ 


.28066 
.28086 
.28106 
.28127 
•28147 


.39016 
.39055 
.39095 
.39134 
•39173 


•29289 
•29310 
•29330 
•29351 
•29372 


.41421 
.41463 
.41504 
.41545 
.41586 


.30534 
.30555 
.30576 
•30597 
•30618 


.43956 
.43999 
.44042 
.44086 
•44129 


.31800 
.31821 
.31843 
•31864 
■31885 


•46628 
.46674 
.46719 
.46765 
.46811 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.28167 
.28187 
.28208 
.28228 
•28248 


.39212 
•39251 
•39291 
.39330 
•39369 


.29392 
•29413 
.29433 
•29454 
•29475 


.41627 
•41669 
.41710 
•41752 
.41793 


.30639 
.30660 
.30681 
.30702 
.30723 


•44173 
.44217 
.44260 
•44304 
.44347 


.31907 
•31928 
.31949 
•31971 
.31992 


.46857 
.46903 
.46949 
.46995 
47041 

.47087 
.47134 
.47180 
.47226 
.47272 


5 
6 
7 
8 

q 


10 

11 
12 
13 
14 


.28268 
.28289 
•28309 
.28329 
•28350 


.39409 
.39448 
.39487 
.39527 
.39566 


•29495 
•29516 
.29537 
•29557 
•29578 


.41835 
.41876 
.41918 
.41959 
•42001 


•30744 
.30765 
.30786 
•30807 
•30828 


.44391 
.44435 
.44479 
.44523 
.44567 


.32013 
•32035 
•32056 
•32077 
32099 


10 

11 

12 
13 
14 


15 
16 
17 
18 

19 


.28370 
.28390 
.28410 
•28431 
•28451 


.39606 
•39646 
.39685 
.39725 
•39764 


.29599 
.29619 
.29640 
•29661 
•29681, 


.42042 
.42084 
.42126 
•42168 
42210 


.30849 
.30870 
.30891 
30912 
30933 


•44610 
.44654 
.44698 
. 44742 
•44787 


•32120 
•32141 
•32163 
•32184 
32205 


.47319 
.47365 
.47411 
.47458 
■47504 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


•28471 
•28492 
•28512 
•28532 
.28553 


•39804 
•39844 
.39884 
.39924 
•39963 


.29702 
.29723 
.29743 
.29764 
•29785 


.42251 
.42293 
.42335 
.42377 
•42419 


.30954 
•30975 
•30996 
•31017 
31038 


.44831 
•44875 
.44919 
.44963 
•45007 


•32227 
.32248 
.32270 
.32291 
•32312 


.47551 
.47598 
.47644 
.47691 
47738 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


-28573 
.28593 
•28614 
.28634 
•28655 


.40003 
.40043 
.40083 
.40123 
.40163 


29805 
•29826 
•29847 
•29868 

29888 


.42461 
.42503 
.42545 
.42587 
.42630 


31059 
•31080 

31101 
•31122 

31143 


.45052 
.45096 
.45141 
.45185 
.45229 


.32334 
.32355 
•32377 
.32398 
•32420 


.47784 
.47831 
.47878 
.47925 
.47972 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•28675 
.28695 
.28716 
.28736 
.28757 


.40203 
.40243 
.40283 
•40324 
.40364 

.40404 
. 40444 
.40485 
•40525 
.40565 


.29909 
•29930 
•29951 
.29971 
•29992 

•30013 
•30034 
•30054 
•30075 
•30096 


.42672 
.42714 
.42756 
.42799 
.42841 


•31165 
•31186 
31207 
31228 
•31249 


•45274 
.45319 
•45363 
•45408 
•45452 


•32441 
•32462 
•32484 
.32505 
.32527 


.48019 
.48066 
•48113 
.48160 
•48207 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•28777 
•28797 
•28818 
.28838 
.28859 


.42883 
.42926 
.42968 
.43011 
•43053 


•31270 
31291 
31312 
•31334 
■31355 


•45497 
•45542 
•45587 
.45631 
•45676 
.45721 
.45766 
.45811 
.45856 
•45901 


.32548 
•32570 
•32591 
•32613 
■32634 
•32656 
32677 
•32699 
•32720 
•32742 


.48254 
•48301 
.48349 
.48396 
•48443 
.48491 
.48538 
.48586 
.48633 
•48681 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•28879 
•28900 
•28920 
.28941 
.28961 


.40606 
•40646 
.40687 
.40727 
.40768 


•30117 
•30138 
•30158 
.30179 
.30200 


.43096 
.43139 
.43181 
.43224 
.43267 


•31376 
■31397 
•31418 
.31439 
.31461 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•28981 
•29002 
•29022 
•29043 
•29063 
•29084 
.29104 
•29125 
•29145 
•29166 


.40808 
.40849 
.40890 
.40930 
.40971 


•30221 
.30242 
•30263 
•30283 
•30304 


•43310 
•43352 
.43395 
.43438 
-43481 


•31482 
•31503 
.31524 
.31545 
•31567 


•45946 
•45992 
.46037 
.46082 
-46127 


•32763 
32785 
32806 

■32828 
32849 


•48728 
.48776 
48824 
48871 
•48919 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.41012 
•41053 
.41093 
•41134 
.41175 


30325 
30346 
30367 
30388 
30409 


.43524 
.43567 
•43610 
•43653 
•43696 


.31588 
•31609 
•31630 
•31651 
•31673 


.46173 
•46218 
•46263 
.46309 
.46354 


•32871 
.32893 
.32914 
•32936 
32957 


.48967 
•49015 
.49063 
.49111 
-49159 


50 

51 

52 
53 
54 


55 
56 
57 
58 
59 


•29187 
•29207 
29228 
29248 : 
29269 


.41216 
•41257 
.41298 
•41339 
.41380 


30430 
30451 
30471 
30492 
30513 


.43739 
•43783 
43826 
•43869 
.43912 


.31694 
•31715 
•31736 
•31758 
•31779 


.46400 
-46445 
.46491 
.46537 
.46582 


•32979 
33001 
33022 
33044 
33065 


.49207 
.49255 
-49303 
.49351 
-4P399 1 


55 
56 
57 
58 
59 


60 


.29289 j .41421 j 


30534 .43956 1 


31800 


.46628 


33087 .49448 


60 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANT& 
48° 49° 50° 5 J 



t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 





.33087 


.49448 


.34394 


.52425 


.35721 


•55572 


.37068 


•58902 





1 


.33109 


.49496 


.34416 


.52476 


.35744 


•55626 


.37091 


.58959 


i 


9, 


.33130 


.49544 


.34438 


•52527 


.35766 


•55680 


.37113 


•59016 


2 


3 


.33152 


.49593 


.34460 


.52579 


•35788 


•55734 


.37136 


.59073 


3 


4 


.33173 


.49641 


•34482 


•52630 


•35810 


.55789 


.37158 


.59130 


4 


5 


.33195 


.49690 


.34504 


•52681 


•35833 


.55843 


.37181 


.59188 


5 


6 


•33217 


.49738 


•34526 


.52732 


•35855 


.55897 


.37204 


.59245 


6 


7 


33238 


.49787 


•34548 


.52784 


•35877 


.55951 


.37226 


.59302 


7 


8 


•33260 


.49835 


•34570 


.52835 


•35900 


.56005 


.37249 


.59360 


8 


9 


•33282 


.49884 


.34592 


•52886 
.52938 


•35922 


56060 


.37272 


■59418 


9 
10 


10 


•33303 


.49933 


•34614 


•35944 


.56114 


.37294 


.59475 


u 


.33325 


.49981 


.34636 


.52989 


35967 


.56169 


•37317 


.59533 


11 


12 


•33347 


.50030 


•34658 


.53041 


.35989 


.56223 


•37340 


.59590 


12. 


U 


.33368 


.50079 


•34680 


•53092 


•36011 


.56278 


.37362 


.59648 


13 


14 


.33390 


.50128 


•34702 


.53144 


•36034 


•56332 


.37385 


.59706 


14 


L5 


.33412 


.50177 


.34724 


.53196 


.36056 


•56387 


.37408 


•59764 


15 


16 


•33434 


.50226 


.34746 


•53247 


•36078 


. 56442 


•37430 


•59822 


16 


17 


•33455 


.50275 


•34768 


•53299 


•36101 


.56497 


•37453 


•59880 


17 


18 


•33477 


.50324 


.34790 


•53351 


•36123 


.56551 


.37476 


•59938 


18 


19 


•33499 


•50373 


.34812 


•53403 


•36146 


.56606 


.37498 


•59996 


19' 


20 


•33520 


•50422 


.34834 


•53455 


•36168 


.56661 


.37521 


•60054 


30 


21 


■33542 


.50471 


•34856 


•53507 


•36190 


.56716 


.37544 


•60112 


21 


22 


.33564 


.50521 


•34878 


•53559 


•36213 


•56771 


.37567 


•60171 


22 


23 


.33586 


.50570 


.34900 


•53611 


•36235 


•56826 


.37589 


•60229 


23 


24 


.33607 


.50619 


.34923 


•53663 


•36258 


.56881 


■37612 


.60287 


24 


25 


.33629 


.50669 


.34945 


.53715 


•36280 


•56937 


•37635 


•60346 


25 


26 


•33651 


•50718 


•34967 


•53768 


•36302 


.56992 


•37658 


•60404 


26 


27 


.33673 


•50767 


•34989 


•53820 


•36325 


.57047 


•37680 


•60463 


27 


28 


.33694 


.50817 


.35011 


.53872 


.36347 


.57103 


•37703 


.60521 


28 


29 


.33716 


•50866 


.35033 


.53924 


•36370 


•57158 


•37726 


•60580 


29' 


30 


.33738 


.50916 


•35055 


.53977 


•36392 


•57213 


•37749 


.60639 


301 


31 


.33760 


.50966 


.35077 


•54029 


.36415 


.57269 


•37771 


.60698 


31 


32 


.33782 


.51015 


.35099 


•54082 


•36437 


•57324 


•37794 


.60756 


32 


33 


.33803 


.51065 


•35122 


•54134 


•36460 


•57380 


•37817 


•60815 


33 


34 


•33825 


• 51115 


.35144 


.54187 


36482 


•57436 


.37840 


•60874 


34 
35 


35 


•33847 


•51165 


.35166 


. 54240 


-36504 


•57491 


.37862 


.60933 


36 


.33869 


.51215 


.35188 


.54292 


•36527 


•57547 


•37885 


•60992 


36 


37 


33891 


.51265 


•35210 


•54345 


•36549 


.57603 


•37908 


•61051 


37 


38 


•33912 


•51314 


•35232 


•54398 


•36572 


.57659 


•37931 


.61111 


38 


39 


.33934 


.51364 


•35254 


.54451 


.36594 
•36617 


•57715 
.57771 


.37954 
•37976 


•61170 
•61229 


39 


40 


•33956 


.51415 


.35277 


.54504 


40 


41 


•33978 


.51465 


.35299 


.54557 


■36639 


.57827 


.37999 


.61288 


41 


42 


•34000 


•51515 


.35321 


•54610 


•36662 


.57883 


•38022 


•61348 


42: 


43 


•34022 


•51565 


•35343 


•54663 


•36684 


.57939 


•38045 


.61407 


43-: 


44 


•34044 


•51fil5 


•35365 


.54716 


•36707 


.57995 


■38068 
•38091 


•61467 


44 


45 


•34065 


.51665 


.35388 


.54769 


•36729 


•58051 


.61526 


45 


46 


•34087 


•51716 


.35410 


•54822 


•36752 


.58108 


•38113 


•61586 


46 


47 


•34109 


•51766 


.35432 


.54876 


•36775 


.58164 


•38136 


.61646 


47 


48 


•34131 


•51817 


•35454 


.54929 


•36797 


•58221 


•38159 


.61705 


48 


49 


•34153 


.51867 
•51918 


•35476 


.54982 


•36820 


•58277 


•38182 


•61765 


49 
50 


50 


•34175 


•35499 


.55036 


•36842 


.58333 


•38205 


.61825 


51 


•34197 


•51968 


•35521 


.55089 


•36865 


• .58390 


•38228 


.61885 


51 


52 


•34219 


.52019 


•35543 


.55143 


•36887 


.58447 


■38251 


.61945 


52. 


53 


•34241 


.52069 


•35565 


.55196 


•36910 


.58503 


•38274 


.62005 


53 


54 


•34262 


.52120 


.35588 


.55250 


•36932 


.58560 


38296 


.62065 


54 


55 


•34284 


•52171 


.35610 


.55303 


.36955 


.58617 


•38319 


.62125 


55 


56 


•34306 


•52222 


•35632 


•55357 


.36978 


.58674 


-38342 


.62185 


56 


57 


•34328 


.52273 


•35654 


.55411 


•37000 


•58731 


.38365 


.62246 


57 


58 


34350 


.52323 


.35677 


.55465 


•37023 


•58788 


.38388 


.62306 


58 


59 


.34372 


.52374 


•35699 


.55518 


■37045 


.58845 


•38411 


.62366 


59 


60 


134394 


•52425 


•35721 


.55572 


•37068 


•58902 


.38434 


•62427 


60 



718 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
53° 53° 54° SS*" 





1 

2 
3 
4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 




.38434 
.38457 
.38480 
.38503 
.38526 

•38549 
.38571 
.38594 
•38617 
•38640 


•62427 
.62487 
.62548 
.62609 
_62_669^ 
.62730 
.62791 
.62852 
.62913 
•62974 


.39819 
•39842 
•39865 
.39888 
^9_^911 
.39935 
•39958 
•39981 
•40005 
.40028 


.66164 
.66228 
.66292 
•66357 
•66421 


.41221 
•41245 
.41269 
.41292 
•41316 


.70130 
.70198 

.70267 
.70335 
• 70403 


•42642 
•42666 
•42690 
•42714 
•42738 


. 74345 
.74417 
. 74490 
.74562 
.74635 




1 
2 
3 
4 


•66486 
•66550 
.66615 
.66679 
•66744 


•41339 
•41363 
•41386 
•41410 
•41433 


. 70472 
.70540 
.70609 
.70677 
. 70746 


•42762 
•42785 
•42809 
•42833 
•42857 


. 74708 
.74781 
•74854 
•74927 
•75000 


5 
6 
7 
8 
9 


•38663 
.38686 
.38709 
.38732 
•38755 


•63035 
.63096 
.63157 
.63218 
•63279 


.40051 
.40074 
•40098 
.40121 
•40144 


.66809 
.66873 
•66938 
.67003 
.67068 


.41457 
•41481 
.41504 
•41528 
.41551 


70815 
. 70884 
.70953 
.71022 
•71091 


•42881 
•42905 
•42929 
•42953 
•42976 


•75073 
.75146 
.75219 
.75293 
•75366 


10 

11 

12 
13 
14 


15 
16 
17 
18 

IL. 
ro 

21 
22 
23 
24 

25 
26 
27 
28 
29 

;o 

n 

J2 
53 
J4 


•38778 
•38801 
•38824 
.38847 
•38870 

•38893 
.38916 
.38939 
.38962 
•38985 


•63341 
.63402 
.63464 
.63525 
•63587 
•63648 
.63710 
.63772 
•63834 
.63895 

.63957 
.64019 
.64081 
.64144 
.64206 


•40168 
.40191 
•40214 
.40237 
.40261 


.67133 
.67199 
.67264 
.67329 
.67394 


•41575 
.41599 
.41622 
•41646 
.41670 


.71160 
.71229 
.71298 
.71368 
.71437 


.43000 
.43024 
•43048 
43072 
• 4309,6. 


. 75440 
.75513 
.75587 
.75661 
•75734 


15 
16 
17 
18 
-19 


.40284 
.40307 
.40331 
.40354 
.40378 


•67460 
.67525 
.67591 
.67656 
.67722 


.41693 
•41717 
•41740 
.41764 
.41788 

•41811 
.41835 
.41859 
.41882 
.41906 


.71506 
.71576 
.71646 
.71715 
•71785 


•43120 
•43144 
■43168 
.43192 
43216 


•75808 
.75882 
.75956 
•76031 
•76105 


20 

21 
22 
23 
24 


•39009 
•39032 
•39055 
•39078 
•39101 


.40401 
.40424 
.40448 
.40471 
.40494 


.67788 
.67853 
.67919 
.67985 
.68051 


.71855 
.71925 
.71995 
.72065 
•72135 


•43240 
43264 
•43287 
•43311 
43335 


•76179 
•76253 
•76328 
.76402 
•76477 


25 
26 
27 
28 
29 


.39124 
.39147 
.39170 
•39193 
•39216 


.64268 
.64330 
.64393 
.64455 
.64518 


.40518 
.40541 
.40565 
.40588 
•40611 

.40635 
.40658 
.40682 
.40705 
•40728 


.68117 
.68183 
.68250 
.68316 
•68382 


.41930 
.41953 
.41977 
.42001 
• 42024 


.72205 
.72275 
.72346 
.72416 
.72487 


.43359 
.43383 
.43407 
43431 
.43455 


.76552 
.76626 
.76701 
.76776 
.76851 


30 

31 
32 
33 
34 


55 
56 
57 
58 



11 
[2 
t3 
t4 
15 
t6 
t7 
L8 
t9 



1 )1 

2)2 

3 )3 

4 14 

5)5 

6 )6 
I )7 

8 )8 

9 i9 


•39239 
•39262 
•39286 
•39309 
.39332 


.64580 
.64643 
.64705 
.64768 
.64831 


. 68449 
.68515 
.68582 
.68648 
.68715 


•42048 
•42072 
•42096 
•42119 
•42143 


•72557 
.72628 
.72698 
.72769 
• 72840 


•43479 
•43503 
•43527 
•43551 
.43575 


•76926 
.77001 
.77077 
.77152 
•77227 


35 
36 
37 
38 
39 


•39355 
•39378 
•39401 
•39424 
•39447 

.39471 
•39494 
•39517 
•39540 
•39563 


.64894 
.64937 
.65020 
.65083 
.65146 

.65209 
.65272 
.65336 
.65399 
•65462 


.40752 
.40775 
.40799 
.40822 
•40846 


•68782 
.68848 
.68915 
.68982 
.69049 


•42167 
.42191 
.42214 
.42238 
•42262 


.72911 
.72982 
.73053 
.73124 
.73195 


.43599 
•43623 
•43647 
•43671 
.43695 


.77303 
.77378 
.77454 
.77530 
•77606 


40 

41 
42 
43 
44 


.40869 
•40893 
•40916 
•40939 
.40963 


.69116 
.69183 
.69250 
.69318 
.69385 


•42285 
.42309 
.42333 
.42357 
.42381 


.73267 
.73338 
. 73409 
.73481 
•73552 


•43720 
.43744 
.43768 
•43792 
•43816 


.77681 
.77757 
.77833 
.77910 
•77986 


45 
46 
47 
48 
49 


•39586 
.39610 
•39633 
.39656 
.39679 
•39702 
.39726 
•39749 
•39772 
•39795 


•65526 
•65589 
.65653 
.65717 
•65780 


.40986 
•41010 
•41033 
•41057 
•41080 


.69452 
.69520 
.69587 
.69655 
•69723 


.42404 
.42428 
.42452 
.42476 
.42499 


.73624 
.73696 
.73768 
.73840 
•73911 


•43840 
.43864 
.43888 
.43912 
.43936 


.78062 
.78138 
.78215 
.78291 
.78368 

. 78445 
.78521 
.78598 
.78675 
•78752 


50 

51 
52 
53 
54 


.65844 
.65908 
.©5972 
.66036 
.66100 


.41104 
.41127 
.41151 
.41174 
.41198 


.69790 
.69858 
.69926 
.69994 
.70062 


.42523 
.42547 
.42571 
.42595 
•42619 


.73983 
.74056 
.74128 
.74200 
•74272 


•43960 
•43984 
•44008 
.44032 
•44057 


55 
56 
57 
58 
59 


,0,0 .39819 


.66164 


.41221 


.70130 


.42642 


.74345 


.44081 


.78829 


60 




719 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
56° 57° 58° 59° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.44081 
.44105 
.44129 
•44153 
■44177 




78829 
78906 
78984 
79061 
79138 


•45536 
•45560 
•45585 
•45609 
•45634 




83608 
83690 
83773 
83855 
83938 


47008 
•47033 
■47057 
•47082 
•47107 




88708 
88796 
88884 
88972 
89060 


•48496 
•48521 
•48546 
48571 
■48596 


•94160 
.94254 
.94349 
.94443 
.94537 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.44201 
.44225 
•44250 
.44274 
•44298 




79216 
79293 
79371 
79449 
79527 


45658 
■45683 
•45707 

45731 
.45756 




84020 
84103 
84186 
84269 
84352 


•47131 
•47156 
•47181 
•47206 
•47230 




89148 
89237 
89325 
89414 
89503 


•48621 
48646 
•48671 
•48696 
•48721 


.94632 
.94726 
•94821 
.94916 
.95011 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•44322 
•44346 
•44370 
•44395 
•44419 




79604 
79682 
79761 
79839 
79917 


•45780 
45805 
•45829 
•45854 
•45878 




84435 
84518 
84601 
84685 
84768 


•47255 
•47280 
•47304 
.47329 
.47354 




89591 
89680 
89769 
89858 
89948 


•48746 
•48771 
•48796 
•48821 
•48846 


.95106 
.95201 
.95296 
.95392 
•95487 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


• 44443 
•44467 
•44491 
•44516 
•44540 




79995 
80074 
80152 
80231 
80309 


•45903 
•45927 
•45951 
•45976 
•46000 




84852 
84935 
85019 
85103 
85187 


.47379 
•47403 
•47428 
•47453 
•47478 




90037 
90126 
90216 
90305 
90395 


•48871 
.48896 
.48921 
•48946 
•48971 


.95583 
.95678 
.95774 
.95870 
.95966 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


•44564 
44588 

• 44612 
44637 
44661 




80388 
80467 
80546 
80625 
80704 


•46025 
•46049 
•46074 
•46098 
•46123 




85271 
85355 
85439 
85523 
85608 


47502 
•47527 

47552 
•47577 
•47601 




90485 
90575 
90665 
90755 
90845 


•48996 
•49021 
•49046 
•49071 
.49096 


•96062 
.96158 
.96255 
.96351 
•96448 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•44685 
•44709 
•44734 
44758 
•44782 




80783 
80862 
80942 
81021 
81101 


•46147 
•46172 
•46196 
•46221 
•46246 




85692 
85777 
85861 
85946 
86031 


•47626 
•47651 
•47676 
•47701 
•47725 




90935 
91026 
91116 
91207 
91297 


•49121 
•49146 
•49171 
49196 
■49221 


.96544 
.96641 
.96738 
.96835 
.96932 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•44806 
•44831 
• 44855 
44879 
•44903 




81180 
81260 
81340 
81419 
81499 


46270 
•46295 
•46319 
•46344 
•46368 




86116 
86201 
86286 
86371 
86457 


•47750 
•47775 
•47800 
.47825 
•47849 




91388 
91479 
91570 
91661 
91752 


•49246 
•49271 
.49296 
.49321 
.49346 


.97029 
.97127 
.97224 
.97322 
•97420 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•44928 
•44952 
•44976 
45001 
•45025 




81579 
81659 
81740 
81820 
81900 


•46393 
•46417 
•46442 
•46466 
•46491 




86542 
86627 
86713 
86799 
86885 


•47874 
•47899 
•47924 
•47949 
.47974 




91844 
91935 
92027 
92118 
92210 


•49372 
•49397 
•49422 
•49447 
•49472 


.97517 
•97615 
•97713 
.97811 
.97910 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•45049 
•45073 
45098 
•45122 
•45146 




81981 
82061 
82142 
82222 
82303 


•46516 
•46540 
•46565 
•46589 
•46614 




86970 
87056 
87142 
87229 
87315 


.47998 
.48023 
•48048 
•48073 
.48098 




92302 
92394 
92486 
92578 
92670 


•49497 
•49522 
•49547 
•49572 
.49597 


.98008 
.98107 
.98205 
.98304 
.98403 


40 

41 
42 
43. 
44 1 


45 
46 
47 
48 
49 


•45171 
•45195 
•45219 
•45244 
•45268 




82384 
82465 
82546 
82627 
82709 


•46639 
46663 
•46688 
•46712 
•46737 




87401 
87488 
87574 
87661 
87748 


•48123 
.48148 
•48172 
•48197 
.48222 




92762 
92855 
92947 
93040 
93133 


.49623 
.49648 
•49673 
•49698 
•49723 


.98502 
.98601 
.98700 
.98799 
.98899 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•45292 
•45317 
•45341 
45365 
45390 




82790 
82871 
82953 
83034 
83116 


46762 
46786 
•46811 
•46836 
•46860 




87834 
87921 
88008 
88095 
88183 


•48247 
•48272 
.48297 
•48322 
•48347 




93226 
93319 
93412 
93505 
93598 


•49748 
•49773 
•49799 
•49824 
•49849 


.98998 
.99098 
.99198 
.99298 
.99398 


50 

51 

52 . 

53 . 

54 1 


55 
56 
57 
58 
59 


•45414 
•45439 
•45463 
•45487 
■45512 




83198 
83280 
83362 
83444 
83526 


■46885 
•46909 
•46934 
•46959 
.46983 




88270 
88357 
88445 
88532 
88620 


•48372 
.48396 
.48421 
.48446 
•48471 




93692 
93785 
93879 
93973 
94066 


.49874 
•49899 
.49924 
.49950 
•49975 


.99498 
.99598 
.99698 
.99799 
•99899 


55 I 

56 

57 

58 

59 


60 


•45536 




83608 


.47008 




88708 


•48496 




94160 


.50000 |1. 00000 


60 



720 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
60° 61° 63° 63° 





Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


' 




1 

2 
3 
4 


.50000 
.50025 
•50050 
•50076 
•50101 




•00000 
•00101 
•00202 
•00303 
00404 


•51519 
•51544 
51570 
•51595 
•51621 


1.06267 
106375 
1^ 06483 
1^06592 
1^06701 


•53053 
•53079 
■53104 
■53130 
•53156 


1.13005 
1.13122 
1.13239 
1.13356 
1.13473 


.54601 
.54627 
•54653 
•54679 
•54705 


1.20269 
1.20395 
1.20521 
1.20647 
1.20773 




1 
2 
3 

4 


5 
6 
7 
8 
9 


•50126 
•50151 
•50176 
•50202 
•50227 




00505 
00607 
00708 
00810 
00912 


51646 
•51672 
•51697 
•51723 
•51748 


1^06809 
1^06918 
1-07027 
1^07137 
1^07246 


•53181 
.53207 
•53233 
•53258 
•53284 


1.13590 
1.13707 
1.13825 
1.13942 
1.14060 


•54731 
.54757 
•54782 
•54808 
•54834 


11.20900 
1.21026 
1.21153 
1.21280 
1.21407 
1.21535 
1.21662 
1.21790 
1.21918 
1.22045 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•50252 
•50277 
•50303 
•50328 
50353 




01014 
01116 
01218 
01320 
01422 


•51774 
•51799 
•51825 
•51850 
•51876 


1.07356 
1.07465 
1.07575 
1-07685 
1^07795 


•53310 
.53336 
.53361 
53387 
•53413 


1.14178 
1.14296 
1.14414 
1.14533 
1.14651 


•54860 
•54886 
•54912 
•54938 
•54964 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 

•^0 
21 
22 
23 
24 
25 
26 

,27 
28 

l29 


•50378 
. 50404 
■ 50429 
. 50454 
• 50479 




01525 
01628 
01730 
01833 
01936 


•51901 
•51927 
.51952 
51978 
.52003 


1.07905 
1.08015 
1.08126 
1.08236 
1.08347 


.53439 
•53464 
•53490 
-53516 
•53542 


1.14770 
1.14889 
1-15008 
1.15127 
1-15246 


•54990 
■55016 
•55042 
•55068 
55094 


1-22174 
1.22302 
1-22430 
1-22559 
1-22688 


15 
16 
17 
18 
19 


•50505 
•50530 
•50555 
•50581 
•50606 

50631 
-50656 
•50682 
•50707 
•50732 




02039 
02143 
02246 
02349 
02453 
02557 
02661 
02765 
02869 
02973 


•52029 
•52054 
•52080 
•52105 
•52131 

•52156 
•52182 
•52207 
•52233 
•52259 


1-08458 
1-08569 
1.08680 
1.08791 
1-08903 


-53567 
-53593 
-53619 
-53645 
-53670 


1-15366 
1-15485 
1-15605 
1-15725 
1-15845 


■55120 
55146 

■55172 
55198 

.55224 


1.22817 
1-22946 
1- 23075 
1-23205 
1.23334 


20 

21 
22 
23 
24 


1^09014 
1^09126 
1.09238 
1.09350 
1-09462 


•53696 
•53722 
-53748 
-53774 
-53799 


1-15965 
1.16085 
1.16206 
1.16326 
1.16447 


55250 
•55276 
•55302 
.55328 

55354 


1-23464 
1-23594 
1-23724 
1^ 23855 
1. 23985 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•50758 

50783 

50808 

•50834 

•50859 




03077 
03182 
03286 
03391 
03496 


.52284 
•52310 
•52335 
•52361 
•52386 


1^09574 
1.09686 
1.09799 
1^09911 
1-10024 


.53825 
-53851 
•53877 
•53903 
.53928 


1.16568 
1.16689 
1.16810 
1.16932 
1.17053 


55380 
•55406 
•55432 
•55458 
•55484 


1-24116 
1-24247 
1-24378 
1-24509 
1.24640 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•50884 
50910 

•50935 
50960 

•50986 




03601 
03706 
03811 
03916 
04022 


■52412 
•52438 
•52463 
•52489 
•52514 


1.10137 
1.10250 
1.10363 
1.10477 
1.10590 


-53954 
-53980 
- 54006 
-54032 
.54058 


1.17175 
1.17297 
1.17419 
1.17541 
1.17663 


•55510 
•55536 
•55563 
•55589 
55615 


1-24772 
1.24903 
1.25035 
1-25167 
1 25300 


35 
36 
37 
38 
39 


10 

41 
42 
43 
44 

45 
46 
47 
48 
49 


51011 
51036 
51062 
51087 
51113 




04128 
04233 
04339 
04445 
04551 


•52540 
•52566 
•52591 
■52617 
•52642 


1.10704 
1.10817 
1.10931 
1.11045 
1.11159 


- 54083 

- 54109 
•54135 
.54161 
.54187 


1.17786 
1.17909 
1.18031 
1.18154 
1-18277 


•55641 
•55667 
•55693 
•55719 
•55745 


1.25432 
1-25565 
1-25697 
1-25830 
1-25963 


40 

41 
42 
43 
44 


51138 
51163 
51189 
51214 
51239 




04658 
04764 
04870 
04977 
05084 


■52668 
•52694 
52719 
52745 
52771 


1.11274 
1.11388 
1.11503 
1.11617 
1.11732 


.54213 
.54238 
• 54264 
•54290 
•54316 


1.18401 
1.18524 
1.18648 
1.18772 
1. 18895 


55771 
•55797 
•55823 
.55849 
.55876 


1-26097 
1-26230 
1-26364 
1- 26498 
1.26632 


45 
46 
47 
48 
49 


'>0 

51 
52 
53 
54 


51265 
51290 
51316 
51341 
51366 




05191 
05298 
05405 
05512 
05619 


52796 
52822 
52848 
52873 
52899 


1.11847 
1.11963 
1.12078 
1. 12193 
1.12309 


•54342 
•54368 
•54394 
• 54420 
54446 
54471 
54497 
54523 
54549 
54575 


1^19019 
1^19144 
1.19268 
1.19393 
1.19517 


.55902 
.55928 
55954 
55980 
56006 


1-26766 
1-26900 
1-27035 
1.27169 
1.27304 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


51392 
•51417 
•51443 
•51468 

51494 




05727 
05835 
05942 
06050 
06158 


52924 
52950 
52976 
53001 
53027 


1.12425 
L 12540 
1.12657 
1.12773 
1.12889 


1.19642 
1.19767 
1.19892 
1.20018 
1-20143 


56032 
56058 
56084 
56111 
56137 


1-27439 
1-27574 
1-27710 
1-27845 
1- 27981 


55 
56 
57 
58 

5r 


♦0 


•51519 


1. 


06267 


53053 1.13005 | 


54601 


1.20269 


56163 


L. 28117 


60 


721 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
64° 65° 66° 67° 



/ 


Vers. 


Ex. sec. 


Vei-s. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


, 




1 

2 
3 
4 


.56163 
.56189 
.56215 
.56241 
.56267 




.28117 
28253 
28390 
28526 
28663 


.57738 
.57765 
.57791 
.57817 
.57844 


^ 


.36620 
.36768 
•36916 
•37064 
•37212 


•59326 
•59353 
•59379 
•59406 
•59433 




45859 
46020 
46181 
46342 
46504 


•60927 
•60954 
.60980 
.61007 
.61034 




55930 
56106 
56282 
56458 
56634 




1 

2 

1 


5 
6 
7 
8 
9 


.56294 
•56320 
•56346 
•56372 
•56398 




28800 
28937 
29074 
29211 
29349 


•57870 
.57896 
.57923 
•57949 
•57976 


37361 
.37509 
.37658 

37808 
.37957 


•59459 
•59486 
•59512 
•59539 
•59566 




46665 
46827 
46989 
47152 
47314 


.61061 
•61088 
.61114 
.61141 
.61168 




56811 
56988 
57165 
57342 
57520 


5 
6 
? 

8 
9 


10 

11 
12 
13 
14 


•56425 
•56451 
•56477 
•56503 
.56529 




29487 
29625 
29763 
29901 
30040 


•58002 
58028 
.58055 
.58081 
.58108 




38107 
38256 
38406 
38556 
38707 


•59592 
■59619 
.59645 
•59672 
.59699 
.59725 
.59752 
59779 
59805 
59832 




47477 
47640 
47804 
47967 
.48131 


.61195 
•61222 
•61248 
•61275 
.61302 


1 


57698 
57876 
58054 
58233 
58412 


10 

11 

12 
13 
14 


15 
16 
17 
18 
,19 


.56555 
.56582 
.56608 
.56634 
56660 




30179 
30318 
30457 
30596 
30735 


.58134 

.58160 

58187 

58213 

58240 




38857 
39008 
39159 
39311 
39462 




48295 
48459 
48624 
48789 
48954 


.61329 
-61356 
•61383 
•61409 
.61436 




58591 
58771 
58950 
59130 
59311 


15 
16 
17 
18 
19 


20 

21 
C2 
23 
24 


56687 
56713 
56739 
56765 
56791 




30875 
31015 
31155 
31295 
31436 


58266 
58293 
58319 
58345 
58372 




39614 
39766 
39918 
40070 
40222 


59859 
59885 
59912 
59938 
•59965 




49119 
49284 
49450 
49616 
49782 


•61463 
.61490 
.61517 
•61544 
•61570 




59491 
59672 
59853 
60035 
60217 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


56818 
56844 
56870 
56896 
56923 
56949 
56975 
57001 
57028 
57054 




31576 
31717 
31858 
31999 
32140 


58398 
58425 
58451 
58478 
58504 

58531 
58557 
58584 
58610 
58637 
58663 
58690 
58716 
58743 
58769 




40375 
40528 
40681 
40835 
40988 


59992 
■60018 
.60045 
■60072 
■60098 




49948 
50115 
50282 
50449 
50617 


•61597 
•61624 
•61651 
•61678 
•61705 

.61732 
•61759 
•61785 
•61812 
.61839 


I 


60399 
60581 
60763 
60946 
61129 
61313 
61496 
61680 
61864 
62049 
62234 
62419 
62604 
62790 
62976 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 




32282 
32424 
32566 
32708 
32850 




41142 
41296 
41450 
41605 
41760 


■60125 
■60152 
•60178 
•60205 
■60232 




50784 
50952 
51120 
51289 
51457 


30 

31 
3C 
33 
34 


35 
36 
37 
38 
39 


57080 
57106 
57133 
57159 
57185 




32993 
33135 
33278 

33422 
33565 




41914 
42070 
42225 
42380 
42536 


•60259 
•60285 
•60312 
•60339 
•60365 




51626 
51795 
51965 
52134 
52304 


•61866 
•61893 
.61920 
.61947 
•61974 


35 
36 
37 
38 
-39 


40 

41 
42 
43 
44 


57212 

57238 

•57264 

.57291 

57317 




33708 
33852 
33996 
34140 
34284 


58796 
58822 
58849 
58875 
58902 




42692 
42848 
43005 
43162 
43318 


•60392 
•60419 
■60445 
. 60472 
.60499 




52474 
52645 
52815 
52986 
53157 


.62001 
■62027 
■62054 
•62081 
.62108 




63162 
63348 
63535 
•63722 
63909 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


.57343 
.57369 
•57396 
•57422 
57448 




34429 
34573 
34718 
34863 
35009 


58928 
.58955 
58981 
59008 
.59034 




43476 
43633 
43790 
43948 
44106 


•60526 
.60552 
.60579 
•60606 
•60633 




53329 
53500 
53672 
53845 
54017 


•62135 
62162 
.62189 
•62216 
• 62243 
.62270 
-62297 
-62324 
•62351 
•62378 




•64097 
•64285 
• 64473 
•64662 
64851 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.57475 
57501 
•57527 
•57554 
.57580 




35154 
35300 
35446 
35592 
35738 


59061 
.59087 
•59114 
•59140 
.59167 
•59194 
.59220 
•59247 
•59273 
.59300 




44264 
44423 
44582 
44741 
44900 


•60659 

•60686 

•60713 

. 60740 

^60766_ 

•60793 

•60820 

•60847 

.60873 

•60900 




54190 
54363 
54536 
54709 
54883 




•65040 
.65229 
65419 
65609 
65799 


50 

51 
52 
53 
54 


55 
56 
57 
58 
.59L 


.57606 
.57633 
.57659 
.57685 
.57712 




35885 
36031 
36178 
36325 
36473 




•45059 
•45219 
.45378 
.45539 
•45699 




55057 
55231 
55405 
55580 
55755 


•62405 
•62431 
•62458 
•62485 
■62512 




•65989 
•66180 
•66371 
•66563 
■66755 


55 
56 
57 
58 
59 


60 


•57738 


1 


•36620 


•59326 


1 


.45859 


•60927 


1 


55930 


■62539 


1 


.66947 


60 



722 



TABLE X.— 
68 


NATURAL VERSED SINES AND EXTERNAL 
69° 70° 71' 


SECANTS. 


1 
2 
3 
4 
5 
6 
7 
8 
9 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 


.62539 
.62566 
.62593 
•62620 
•62647 




•66947 
•67139 
•67332 
67525 
67718 


•64163 
•64190 
•64218 
•64245 
•64272 




79043 
79254 
79466 
79679 
79891 


•65798 
•65825 
•65853 
•65880 
•65907 


1 


92380 
92614 
92849 
93083 
93318 


•67443 |2 
•67471 2 
•67498 2 

•67526 2 
•67553 12 


07155 
07415 
07675 
07936 
08197 




1 

2 
3 
4 


•62674 
•62701 
•62728 
•62755 
• 62782 




67911 
68105 
68299 
68494 
68689 


•64299 
•64326 
•64353 
•64381 
• 64408 




80104 
80318 
80531 
80746 
80960 


•65935 
•65962 
•65989 
•66017 
• 66044 


1 


93554 
93790 
94026 
94263 
94500 


•67581 
•67608 
•67636 
•67663 
•67691 


2 
2 

I 

2 


08459 
08721 
08983 
09246 
09510 


5 
6 
7 
8 
9 


10 

11 

12 
13 
14 


•62809 
•62836 
•62863 
•62890 
•62917 




68884 
69079 
69275 
69471 
69667 


•64435 

• 64462 

• 64489 
•64517 

• 64544 




81175 
81390 
81605 
81821 
82037 


66071 
•66099 
•66126 
•66154 
•66181 




94737 
94975 
95213 
95452 
95691 


•67718 
•67746 
•67773 
•67801 
•67829 


2 
2 
2 
2 
2 


09774 
10038 
10303 
10568 
10834 


10 

11 
12 
13 
14 


15 
16 
17 
18 

19 


•62944 
•62971 
•62998 
•63025 
•63052 




69864 
70061 
70258 
70455 
70653 


•64571 
•64598 
•64625 
•64653 
•64680 




82254 
82471 
82688 
82906 
83124 


•66208 
•66236 
•66263 
•66290 
•66318 




95931 
96171 
96411 
96652 
96893 


•67856 
•67884 
•67911 
•67939 
•67966 


2 
2 
2 
2 
2. 


11101 
11367 
11635 
11903 
12171 


15 
16 
17 
18 
-19 


10 

21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 


•63079 
•63106 
•63133 
•63161 
•63188 




70851 
71050 
71249 
71448 
71647 


•64707 
• 64734 
•64761 
•64789 
•64816 




83342 
83561 
83780 
83999 
84219 


•66345 
•66373 
•66400 
•66427 
•66455 


1 


97135 
97377 
97619 
97862 
98106 


•67994 
•68021 
• 68049 
•68077 
•68104 


2 
2 
2 

I 


12440 
12709 
12979 
13249 
13520 


20 

21 
22 
23 
24 


•63215 
•63242 
•63269 
63296 
63323 


, 


71847 
72047 
72247 
72448 
72649 


• 64843 
•64870 

• 64898 
•64925 
•64952 




84439 
84659 
84880 
85102 
85323 


•66482 
•66510 
•66537 
•66564 
•66592 




98349 
98594 
98838 
99083 
99329 


•68132 
•68159 
•68187 
•68214 
•68242 


2 
2 
2 
2 
2 


13791 
14063 
14335 
14608 
14881 


25 
26 
27 
28 
29 


•63350 
•63377 
. 63404 
•63431 
•63458 


1 
I 
1 


72850 
73052 
73254 
73456 
73659 


•64979 
•65007 
•65034 
•65061 
•65088 




85545 
85767 
85990 
86213 
86437 


•66619 
•66647 
•66674 
•66702 
■66729 


2 
2 
2 


99574 
99821 
00067 
00315 
00562 


•68270 
•68297 
•68325 
•68352 
■68380 


2 
2 
2 
2 
2 


15155 
15429 
15704 
15979 
16255 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 
10 
41 
42 
43 
44 
45 
46 
47 
48 
49 


•63485 
•63512 
•63539 
•63566 
•63594 




73862 
74065 
74269 
74473 
74677 


•65116 
•65143 
■65170 
■65197 
•65225 


1 


86661 
86885 
87109 
87334 
87560 


■66756 
66784 
■66811 
■66839 
■66866 


2 
2 
2 
2 
2 


00810 
01059 
01308 
01557 
01807 


•68408 
•68435 
•68463 
•68490 
•68518 


2 
2 
2 
2 
2, 


16531 
16808 
17085 
17363 
17641 


35 
36 
37 
38 
39 


•63621 
•63648 
•63675 
•63702 
•63729 




74881 
75086 
75292 
75497 
75703 


•65252 
■65279 
•65306 
•65334 
•65361 




87785 
88011 
88238 
88465 
88692 


■66894 
66921 
■66949 
■66976 
•67003 

■67031 
•67058 
•67086 
•67113 
67141 


2 
2 
2 
2 
2_ 
2 
2 
2 
2 
2 


02057 
02308 
02559 
02810 
03062 
03315 
03568 
03821 
04075 
04329 


•68546 
68573 
•68601 
■68628 
68656 


2 
2 
2 
2 
2 


17920 
18199 
18479 
18759 
19040 


40 

41 

42 
43 
44 


•63756 
•63783 
63810 
•63838 
•63865 




75909 
76116 
76323 
76530 
76737 


•65388 
•65416 
•65443 
•65470 
•65497 




88920 
89148 
89376 
89605 
89834 


•68684 
•68711 
•68739 
•68767 
•68794 


2 
2 
2 
2 
2 


19322 
19604 
19886 
20169 
20453 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 


63892 
•63919 
•63946 
•63973 
• 64000 

•64027 
•64055 
•64082 
•64109 
64136 




76945 
77154 
77362 
77571 
77780 


•65525 
•65552 
•65579 
•65607 
•65634 




90063 
90293 
90524 
90754 
90986 


■67168 
•67196 
•67223 
•67251 
•67278 


2 
2 
2 
2 
2 


04584 
04839 
05094 
05350 
05607 


•68822 
• 68849 
68877 
•68905 
•68932 


2 
2 
2 
2 
2 


20737 
21021 
21306 
21592 
21878 


50 

51 
52 
53 
54 


1 


77990 
78200 
78410 
78621 
78832 


•65661 
•65689 
•65716 
•65743 
■65771 




91217 
91449 
91681 
91914 
92147 


•67306 
•67333 
•67361 
•67388 
•67416 


2 
2 
2 
2 
2 


05864 
06121 
06379 
06637 
06896 


•68960 
68988 
•69015 
• 69043 
•69071 


2 
2 
2 
2 
2 


22165 
22452 
22740 
23028 
23317 


55 
56 
57 
58 
59 


60 


•64163 


1 


79043 


•65798 il 


92380 


■67443 


2 


07155 


69098 


2 


23607 


60 














72 


3 















TABLE X.— NATURAL VERSED STNES AND EXTERNAL SECANTS. 

73° 73° 74° 75° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


' 




1 

2 
3 
4 


•69098 
.69126 
.69154 
.69181 
.69209 


2 
2 
2 
2 
2 


23607 
23897 
24187 
24478 
24770 


•70763 2 
•70791 2 
•70818 2 
.70846 2 
.70874 2 


•42030 
•42356 
42683 
43010 
43337 


•72436 
•72464 
•72492 
•72520 
•72548 


2.62796 
2.63164 
263533 
2 63903 
2^64274 


•74118 
• 74146 
•74174 
•74202 
•74231 


2 
2 
2 
2 
2 


86370 
86790 
87211 
87633 
88056 




2 
3 
4 


5 
6 

7 
8 
9 


.69237 
.69264 
•69292 
•69320 
69347 


2 
2 
2 

2 
2 


25062 
25355 
25648 
25942 
26237 


.70902 
•70930 
.70958 
.70985 
•71013 


2 
2 
2 
2 
2 


43666 
43995 
44324 
44655 
44986 


•72576 
.72604 
.72632 
.72660 
.72688 


2-64645 
265018 
2.65391 
2-65765 
2.66140 


-74259 
.74287 
.74315 
.74343 
.74371 


2 
2 
2 
2 
2 


88479 
88904 
89330 
89756 
90184 


5 
6 
7 
8 
9 


10 

11 

12 
13 
14 


69375 
•69403 

69430 
•69458 
•69486 


2 
2 
2 
2 

2 


26531 
26827 
27123 
27420 
27717 


.71041 
•71069 
•71097 
•71125 
•71153 


2 
2 

2 
2 
2 


45317 
45650 
45983 
46316 
46651 


.72716 
.72744 
•72772 
•72800 
.72828 


2.66515 
2.66892 
2.67269 
2-67647 
2.68025 


.74399 
.74427 
.74455 
. 74484 
.74512 


2 
2 
2 
2 
2 


90613 
91042 
91473 
91904 
92337 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


•69514 
•69541 
69569 
69597 
•69624 


2 
2 
2 
2 
2 


28015 
28313 
28612 
28912 
29212 


•71180 
•71208 
•71236 
•71264 
.71292 


2 
2 

2 

2 
2 


46986 
47321 
47658 
47995 
48333 


.72856 
.72884 
.72912 
•72940 
•72968 


2^68405 
268785 
2^69167 
2.69549 
2.69931 


•74540 
.74568 
.74596 
.74624 
.74652 


2 
2 
2 
2 
2 


92770 
93204 
93640 
94076 
94514 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


69652 
•69680 
•69708 
.69735 
.69763 


2 
2 
2 
2 

2 


29512 
29814 
30115 
30418 
30721 


•71320 12 
•71348 2. 
•71375 !2. 
•71403 \2 
•71431 !2 


48671 
49010 
49350 
49691 
50032 


•72996 
•73024 
•73052 
.73080 
•73108 


2^70315 
2^70700 
271085 
2^71471 
2-71858 


. 74680 
•74709 
74737 
•74765 
.74793 


2 
2 

2 
2 
2 


94952 
95392 
95832 
96274 
96716 


20 

21 

22 
23 
24 


25 
26 
27 
28 
29 


•69791 
•69818 
•69846 
•69874 
•69902 
•69929 
69957 
•69985 
•70013 
• 70040 


2 

2. 

2. 

2. 

2 


31024 
31328 
31633 
31939 
32244 


•71459 
•71487 
•71515 
.71543 
.71571 


2 
2 
2- 
2. 
2. 


50374 
50716 
51060 
51404 
51748 


•73136 . 
.73164 . 
•73192 : 
• 73220 '. 
.73248 : 


2 • 72246 
2^72635 
2.73024 
2.73414 
2.73806 
2.74198 
2.74591 
2.74984 
2.75379 
2.75775 


.74821 
.74849 
•74878 
.74906 
.74934 


2 
2 
2 
2 
2 


97160 
97604 
98050 
98497 
98944 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


2 
2 
2 
2 
2 


32551 
32858 
33166 
33474 
33783 


.71598 
•71626 
•71654 
•71682 
•71710 


2. 

2. 
2. 
2. 
2 


52094 
52440 
52787 
53134 
53482 

53831 
54181 
54531 
54883 
55235 

55587 
55940 
56294 
56649 
57005 


•73276 i 
73304 : 
73332 : 
73360 : 

.73388 : 


.74962 
•74990 
•75018 
•75047 
.75075 


2 
2 
3 
3 
3 


99393 
99843 
00293 
00745 
01198 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•70068 
•70096 
•70124 
•70151 
70179 


2 
2 
2 
2 
2 


34092 
34403 
34713 
35025 
35336 


•71738 
71766 
•71794 
•71822 
.71850 


2 

2. 
2 
2. 

2. 


.73416 . 

•73444 : 

•73472 : 

•73500 

•73529 

•73557 

•73585 

•73613 

•73641 

•73669 


2.76171 
2.76568 
2.76966 
2.77365 
2.77765 


.75103 
•75131 
•75159 
•75187 
•75216 


3 
3 
3 
3 
3 


01652 
02107 
02563 
03020 
03479 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•70207 
•70235 
•70263 
•70290 
•70318 


2 

2 
2 
2 
2 


35649 
35962 
36276 
36590 
36905 


.71877 
•71905 
•71933 
•71961 
•71989 


2 
2 
2 
2 
2 


2-78166 
2-78568 
2-78970 
2-79374 
2-79778 


•75244 
•75272 
•75300 
•75328 
•75356 


3 
3 
3 
3 
3 


03938 
04398 
04860 
05322 
05786 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


•70346 
•70374 
•70401 
.70429 
•70457 


2 
2 
2 
2 
2 


37221 
37537 
37854 
38171 
38489 


•72017 
•72045 
•72073 
•72101 
.72129 


2. 
2 
2 
2 
2_ 
2 
2 
2 
2 
2 


57361 
57718 
58076 
58434 
58794 


•73697 
•73725 
•73753 
•73781 
•73809 


2-80183 
2-80589 
2-80996 
2-81404 
2. 81813 


•75385 
•75413 
- 75441 
•75469 
.75497 


3 
3 
3 
3 

3 


06251 
06717 
07184 
07652 
08121 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•70485 
•70513 
•70540 
•70568 
.70596 


2 
2 
2 


38808 
39128 
39448 
39768 
40089 


.72157 
.72185 
.72213 
•72241 
.72269 


59154 
59514 
59876 
60238 
60601 


.73837 
•73865 
•73893 
•73921 
•73950 


2-82223 
2-82633 
2-83045 
2-83457 
2-83871 


•75526 
•75554 
•75582 
•75610 
■75639 


3 
3 
3 
3 
3 


08591 
09063 
09535 
10009 
10484 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•70624 
.70652 
.70679 
.70707 
•70735 


2 
2 
2 
2 
2. 


40411 
40734 
41057 
41381 
41705 . 


.72296 
.72324 
.72352 
.72380 
•72408 


2 
2 
2 
2 
2 


60965 
61330 
61695 
62061 
62428 


•73978 
•74006 

• 74034 
•74062 

• 74090 


2-84285 
2-84700 
2-85116 
2^85533 
2^85951 


•75667 
75695 
•75723 
•75751 
•75780 


3 
3 
3 
3 
3 


10960 
11437 
11915 
12394 
12875 


55 
56 
57 
58 
59 


60 


.70763 


2 


.42030 


.72436 


2 


.62796 


•74118 


2 86370 


.75808 


3 


13357 


60 



724 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
76° 77° 78° 79° 



' 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. vsec. 


Vers. 


Ex. sec. 


f 




1 

2 
3 
4 


75808 
•75836 
•75864 
•75892 
•75921 


3 
3 
3 
3 
3 


•13357 
•13839 

•14323 
•14809 
•15295 


•77505 
•77533 
■77562 
•77590 
•77618 


3.44541 
3-45102 
3-45664 
3-46228 
3. 46793 


•79209 
•79237 
•79266 
•79294 
-79323 


3 80973 
381633 
3^82294 
3-82956 
3-83621 


-80919 
-80948 
-80976 
•81005 
•81033 


4-24084 
4-24870 
4-25658 
4-26448 
4^27241 




1 
2 
3 
4 


5 
6 
7 
8 
9 


•75949 
•75977 
•76005 
• 76034 
76062 


3 
3 
3 
3 
3 


•15782 
16271 
16761 
17252 
17744 


•77647 
•77675 
•77703 
•77732 
•77760 


3-47360 
3-47928 
3 48498 
3^49069 
3^49642 


•79351 
•79380 
• 79408 
•79437 
•79465 


3-84288 
3-84956 
3-85627 
3^86299 
3-86973 
3-87649 
3^88327 
3-89007 
3 • 89689 
3 • 90373 


•81062 
•81090 
•81119 
•81148 
81176 


4-28036 
4^28833 
4^29634 
430436 
4. 31241 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•76090 
•76118 
•76147 
•76175 
•76203 


3 
3 
3 
3 
3 


18238 
18733 
19228 
19725 
20224 


•77788 
•77817 
•77845 
•77874 
•77902 


3 • 50216 
3 50791 
3. 51368 
3^51947 
3^52527 


• 79493 
•79522 
•79550 

• 79579 
-79607 


•81205 
•81233 
•81262 
•81290 
81319 


4^32049 
4^32859 
4^33671 
4 •34486 
4^35304 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


•76231 
•76260 
•76288 
•76316 
•76344 


3 
3 
3 
3 
3 


20723 
21224 
21726 
22229 
22734 


•77930 
.77959 
•77987 
•78015 
. 78044 


3-53109 
3.53692 
3 . 54277 
3 . 54863 
3-55451 


-79636 
•79664 
•79693 
•79721 
•79750 


391058 
3 91746 
3 92436 
393128 
3-93821 


•81348 
•81376 
•81405 
•81433 
-81462 


4^36124 
4^36947 
4^37772 
4^38600 
4^39430 


15 
16 
17 
18 
.19 


20 

21 
22 
23 
24 


•76373 
-76401 
•76429 
•76458 
76486 


3 
3 
3 
3 
3_ 

3 
3 
3 
3 
3 


23239 
23746 
24255 
24764 
25275 

.25787 
.26300 
•26814 
•27330 
•27847 


.78072 
.78101 
.78129 
•78157 
-78186 
•78214 
. 78242 
.78271 
•78299 
•78328 


3.56041 
3.56632 
3.57224 
3 • 57819 
3-58414 


•79778 
79807 
•79835 
•79864 
•79892 


3-94517 
3-95215 
3^95914 
3^96616 
3^97320 


•81491 
•81519 
•81548 
•81576 
•81605 


4-40263 
4-41099 
4-41937 
4-42778 
4^43622 


20 

2^ 
21 
23 
24 


25 
26 
27 
28 
29 


•76514 
•76542 
•76571 
•76599 
•76627 


3-59012 
359611 
3 60211 
3-60813 
3- 61417 


•79921 
•79949 
•79978 
•80006 
80035 


3^98025 
3-98733 
3-99443 
4-00155 
4-00869 


•81633 
•81662 
•81691 
•81719 
•81748 


4^44468 
4.45317 
4-46169 
4-47023 
447881 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•76655 
•76684 
•76712 
• 76740 
76769 


3 
3 
3 
3 
3 


•28366 
•28885 
.29406 
•29929 
30452 


•78356 
.78384 
•78413 

• 78441 

• 78470 


3 . 62023 
3^62630 
3.63238 
3 63849 
3 . 64461 


80063 
•80092 
•80120 
•80149 
•80177 


4-01585 
4-02303 
4^03024 
4^03746 
4- 04471 


•81776 
•81805 
•81834 
-81862 
-81891 


4-48740 
4-49603 
4.50468 
4.51337 
4-52208 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


76797 
•76825 
•76854 
•76882 
•76910 


3 
3 
3 
3 
3 


30977 
31503 
32031 
32560 
33090 


. 78498 
.78526 
•78555 
•78583 
•78612 


3^65074 
3-65690 
3-66307 
3-66925 
3. 67545 


•80206 
•80234 
•80263 
•80291 
80320 


405197 
4-05926 
4-06657 
4^07390 
4^08125 


-81919 
-81948 
-81977 
.82005 
•82034 


4.53081 
4.53958 
4.54837 
4.55720 
4.56605 
4.57493 
4.58383 
4.59277 
4.60174 
4-61073 


35 
36 
37 
38 
-39 


40 

41 
42 
43 
44 


•76938 

•76967 

76995 

77023 

77052 


3 
3 
3 
3 
3 


33622 
34154 
34689 
35224 
35761 


•78640 
•78669 
•78697 
• 78725 
•78754 


3 68167 
3.68791 
3.69417 
3 . 70044 
3.70673 


80348 
-80377 

• 80405 

• 80434 
-80462 


4^08863 
4-09602 
4^10344 
4^11088 
4^11835 


.82063 
.82091 
.82120 
.82148 
•82177 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


77080 
77108 
77137 
77165 
77193 


3. 
3 
3^ 
3^ 
3. 


36299 
36839 
37380 
37923 
38466 


•78782 
•78811 
•78839 
•78868 
•78896 


3-71303 
3.71935 
3.72569 
3 • 73205 
3.73843 


-80491 
•80520 
•80548 
•80577 
-80605 


4^12583 
4^13334 
4^14087 
4^14842 
4.15599 


•82206 
•82234 
•82263 
-82292 
-82320 


4.61976 
4.62881 
4-63790 
4^64701 
4^65616 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


77222 
77250 
77278 
•77307 
77335 


3. 
3. 
3. 
3^ 
3 


39012 
39558 
40106 
40656 
41206 


78924 
78953 
78981 
79010 
79038 


3.74482 
3.75123 
3 • 75766 
3 • 76411 
3^77057 


•80634 
•80662 
•80691 
•80719 
•80748 


4^16359 
4.17121 
4.17886 
4.18652 
4-19421 


•82349 
-82377 
-82406 
•82435 
•82463 


4.66533 
4^67454 
468377 
4-69304 
4-70234 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•77363 
•77392 
.77420 
• 77448 
•77477 


3. 

3. 

3 

3 

3 


41759 
42312 
42867 
43424 
43982 


•79067 
•79095 
79123 
•79152 
.79180 
.79209 


3^77705 
3.78355 
3.79007 
3.79661 
3^80316 
3.80973 


80776 
80805 
80833 
80862 
80891 
80919 


4-20193 
4-20966 
4-21742 
4-22521 
4^23301 
4-24084 


•82492 
•82521 
•82549 
•82578 
•82607 


4-71166 
4-72102 
4-73041 
4-73983 
4-74929 


55 
56 
57 
58 
_59 


60 


•77505 


3 


44541 


•82635 


4.75877 


60 



725 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 

80° 81° 83° 83° 



' 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


' 




1 

2 
3 
4 


.82635 
.82664 
.82692 
.82721 
•82750 

82778 
•82807 

82836 
•82864 

82893 


4.75877 
4.76829 
4.77784 
4.78742 
4.79703 


.84357 
-84385 
-84414 
. 84443 
- 84471 


5 
5 
5 
5 

5 


39245 
40422 
41602 
42787 
43977 


•86083 
.86112 
•86140 
.86169 
•86168 
.86227 
•86256 
•86284 
•86313 
•86342 


6 
6 
6 
6 
6 


18530 
20020 
21517 
23019 
24529 


•87813 
.87842 
.87871 
.87900 
-87929 


7 
7 
7 
7 
7 


20551 
22500 
24457 
26425 
28402 




1 
2 
3 
4 


5 
6 
7 
8 
9 


4.80667 
4.81635 
4.82606 
4.83581 
4-84558 


.84500 
.84529 
-84558 
-84586 
-84615 


5 
5 

I 


45171 
46369 
47572 
48779 
49991 


6 
6 
6 
6 
6 


26044 
27566 
29095 
30630 
32171 


-87957 
-87986 
-88015 
-88044 
•88073 


7 
7 
7 
7 
7 


30388 
32384 
34390 
36405 
38431 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


82922 
•82950 
•82979 
83008 
83036 


4.85539 
4-86524 
4.87511 
4.88502 
4.89497 


-84644 
.84673 
•84701 
-84730 
-84759 


5 
5 
5 


51208 
52429 
53655 
54886 
56121 


•86371 
•86400 
86428 
•86457 
•86486 


6 
6 
6 
6 
6 
6 
6 
6 
6 
6 


33719 
35274 
36835 
38403 
39978 


•88102 
•88131 
•88160 
•88188 
•88217 


7 
7 
7 
7 
7 


40466 
42511 
44566 
46632 
-48707 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


83065 
• 83094 

83122 
•83151 

83^8g 

83208 
•83237 
83266 
83294 
83323 


4.90495 
4-91496 
4-92501 
4.93509 
4-94521 
4.95536 
4.96555 
4.97577 
4-98603 
4-99633 


.84788 
•84816 
.84845 
.84874 
.84903 
•84931 
•84960 
.84989 
.85018 
.85046 


5 
5 
5 
5 
5 


57361 
58606 
59855 
61110 
62369 


•86515 
•86544 
•86573 
•86601 
•86630 


41560 
43148 
44743 
46346 
47955 


-88246 
-88275 
•88304 
•88333 
•88362 


7 
7 
7 
7 
7 


50793 

52889 

-54996 

.57113 

59241 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


5 
5. 
5 
5 
5 


63633 
64902 
66176 
67454 
68738 


•86659 
•86688 
•86717 
•86746 
•86774 


6 
6 
6 
6 
6 


49571 
51194 
52825 
54462 
56107 


•88391 
•88420 
•88448' 
•88477 
•88506 
•88535 
•88564 
•88593 
88622 
•88651 


7 
7 
7 
7 
7 


61379 
63528 
65688 
67859 
70041 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


83352 
83380 
83409 
83438 
83467 


5.00666 
5.01703 
5.02743 
5.03787 
5.04834 


-85075 
-85104 
•85133 
85162 
.85190 


5 
5 
5 
5 
5 


70027 
71321 
72620 
73924 
75233 


•86803 
•86832 
•86861 
•86890 
•86919 


6 
6 
6 
6 
6 


57759 
59418 
61085 
62759 
64441 


7 
7 
7 
7 
7 


72234 
74438 
76653 
78880 
81118 


25 
26 
27 
28 
29 


30 

31 
32 
33 

34 


83495 
83524 
83553 
83581 
83610 


5 05886 
5 06941 
5^08000 
5^09062 
5^10129 


-85219 
.85248 
•85277 
•85305 
•85334 


5 
5 
5 
5 
5 


76547 
77866 
79191 
80521 
81856 


•86947 
•86976 
•87005 
•87034 
•87063 


6 
6 
6 
6 
6 


66130 
67826 
69530 
71242 
72962 


•88680 
•88709 
•88737 
•88766 
•88795 


7 
7 
7 
7 
7 


83367 
85628 
87901 
90186 
92482 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


83639 
83667 
83696 
•83725 
83754 


5^11199 
5.12273 
5.13350 
5.14432 
5^15517 


•85363 
.85392 
-85420 
.85449 
.85478 


5 
5 
5 
5 
5 


83196 
84542 
85893 
87250 
88612 


•87092 
-87120 
•87149 
-87178 
-87207 


6 
6 
6 
6 
6 


74689 
76424 
78167 
79918 
81677 


•88824 

88853 

-88882 

•88911 

88940 


7 
7 
7 
8 
8 


94791 
97111 
99444 
01788 
04146 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•83782 
•83811 
•83840 
•83868 
83897 


5^16607 
5^17700 
5.18797 
5.19898 
5.21004 


-85507 
.85536 
•85564 
.85593 
-85622 


I 

5 
5 

5 


89979 
91352 
92731 
94115 
95505 


•87236 
87265 
87294 
87322 

•87351 


6 
6 
6 
6 
6 


83443 
85218 
87001 
88792 
90592 


-88969 
•88998 
•89027 
•89055 
•89084 


8 
8 
8 
8 
8 


06515 
08897 
11292 
13699 
16120 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•83926 
.83954 
.83983 
•84012 
• 84041 


5-22113 
5-23226 
5.24343 
5.25464 
5.26590 


-85651 
85680 
•85708 
-85737 
.85766 


5 
5 
5 
6 
6 


96900 
98301 
99708 
01120 
02538 


•87380 
87409 
87438 
87467 
87496 


6 
6 
6 
6 
6 


92400 
94216 
96040 
97873 
99714 


•89113 
-89142 
-89171 
-89200 
.89229 


8 
8 
8 
8 
8 


18553 
20999 
23459 
25931 
28417 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•84069 
•84098 
•84127 
.84155 
.84184 


5-27719 
5-28853 
5-29991 
5-31133 
5-32279 


-85795 
-85823 
-85852 
-85881 
.85910 


6 
6 
6 
6 
6 


03962 
05392 
06828 
08269 
09717 


•87524 
•87553 
•87582 
•87611 
87640 


7 
7 
7 
7 
7 


01565 
03423 
05291 
07167 
09052 


•89258 
-89287 
-89316 
.89345 
.89374 


8 
8 
8 
8 
8 


30917 
33430 
35957 
38497 
41052 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


.84213 
.84242 
.84270 
.84299 
• 84328, 

.84357 


5.33429 
5.34584 
5.35743 
5.36906 
5 38073 
5.39245 


-85939 
-85967 
-85996 
-86025 
-86054 


6 
6 
6 
6 
6 


11171 
12630 
14096 
15568 
17046 


•87669 
-87698 
-87726 
-87755 
-87784 


7 
7 
7 
7 
7 


10946 
12849 
14760 
16681 
18612 


•89403 
.89431 
-89460 
-89489 
•89518 


8 
8 
8 
8 
8 


43620 
46203 
48800 
51411 
54037 


55 
56 
57 
58 
59 


60 


-86083 


6 


18530 


-87813 


7 


20551 


-89547 


8 


56677 


60 



726 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
84° 85° 86° 



' 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 





•89547 


8.56677 


•91284 


10^47371 


.93024 


13-33559 





i 


•89576 


8.59332 


•91313 


10^51199 


.93053 


13-39547 


1 


2. 


•89605 


8.62002 


•91342 


10^55052 


•93082 


13.45586 


2 


3 


•89634 


8.64687 


•91371 


10.58932 


.93111 


13-51676 


3 


4 


•89663 


8.67387 


-91400 


10.62837 


.93140 
-93169 


13-57817 


4 


5 


•89692 


8.70103 


-91429 


10.66769 


13-64011 


5 


B 


•89721 


8 •72833 


-91458 


10.70728 


•93198 


13.70258 


6 


'/ 


•89750 


8^75579 


•91487 


10^74714 


•93227 


13-76558 


7 


8 


•89779 


8 •78341 


91516 


10.78727 


•93257 


13-82913 


8 


9 


89808 


8.81119 


•91545 


10.82768 


93286 


13-89323 


9 


10 


89836 


8.83912 


91574 


10.86837 


93315 


13-95788 


10 


li 


•89865 


8.86722 


91603 


10^90934 


93344 


14-02310 


11 


12 


•89894 


8.89547 


91632 


10-95060 


93373 


14-08890 


12 


13 


•89923 


8.92389 


91661 


10^99214 


93402 


14-15527 


13 


14 


•89952 


8.95248 


91690 


11.03397 


93431 


14-22223 


14 


15 


•89981 


8-98123 


91719 


a. 07610 


93460 


14^28979 


15 


16 


•90010 


9.01015 


91748 


11.11852 


93489 


14^35795 


16 


17 


•90039 


9.03923 


91777 


11.16125 


93518 


14-42672 


17 


18 


•90068 


9.06849 


91806 


11.20427 


93547 


14.49611 


18 


19 


•90097 
•90126 


9.09792 


91835 


11^24761 


93576 


14-56614 


19 


20 


9.12752 


91864 


11.29125 


93605 


14-63679 


30 


21 


•90155 


9.15730 


91893 


11.33521 


93634 


14-70810 


21 


22 


•90184 


9.18725 


91922 


11.37948 


93663 


14-78005 


22 


23 


90213 


9.21739 


91951 


11.42408 


93692 


14-85268 


23 


24 


•90242 


9.24770 


91980 


11.46900 


93721 


14.92597 


24 


25 


•90271 


9.27819 


92009 


11.51424 


93750 


14.99995 


25 


26 


•90300 


9.30887 


92038 


11.55982 


93779 


15.07462 


26 


27 


•90329 


9.33973 


92067 


11.60572 


93808 


15.14999 


27 


28 


•90358 


9^37077 


92096 


11.65197 


93837 


15.22607 


28 


29 


90386 


9.40201 


92125 


11.69856 


93866 


15-30287 


29 


30 


•90415 


9-43343 


92154 


11.74550 


93895 


15.38041 


30 


31 


• 90444 


9.46505 


92183 


11.79278 


93924 


15-45869 


31 


32 


•90473 


9.49685 


92212 


11.84042 


93953 


15-53772 


32 


33 


•90502 


9-52886 


92241 


11.88841 


93982 


15-61751 


33 


34 


90531 


9.56106 


92270 


11.93677 


94011 


15-69808 


34 


35 


•90560 


9.59346 


92299 


11.98549 


94040 


15-77944 


35 


36 


•90589 


9.62605 


92328 


12 •03458 


94069 


15.86159 


36 


37 


•90618 


9.65885 


92357 


12^08040 


94098 


15.94456 


37 


38 


•90647 


9.69186 


92386 


12.13388 


94127 


16.02835 


38 


39 


•90676 


9.72507 


92415 


12.18411 


94156 


16.11297 


39 


40 


•90705 


9.75849 


92444 


12.23472 


94186 


16-19843 


40 


41 


•90734 


9.79212 


92473 


12.28572 


94215 


16,28476 


41 


42 


•90763 


9-82596 


92502 


12.33712 


94244 


16-37196 


42 


43 


•90792 


9.86001 


92531 


12.38891 


94273 


16-46005 


43 


44 


•90821 


9-89428 


92560 


12.44112 


94302 


16-54903 


44 


45 


•90850 


9-92877 


92589 


12^49373 


94331 


16-63893 


45 


46 


•90879 


9-96348 


92618 


12^ 54676 


94360 


16-72975 


46 


47 


90908 


9-99841 


92647 


12^60021 


94389 


16-82152 


47 


48 


■90937 


10-03356 


92676 


12^65408 


94418 


16-91424 


48 


49 


90966 


10-06894 


92705 


12^ 70838 


94447 


17-00794 


49 


50 


-90995 


10-10455 


92734 


12.76312 


94476 


17-10262 


50 


51 


•91024 


10-14039 


92763 


12.81829 


94505 


17-19830 


51 


62 


•91053 


10-17646 


92792 


12.87391 


94534 


17-29501 


52 


53 


•91082 


10-21277 


92821 


12.92999 


94563 


17-39274 


53 


54 


.91111 


10.24932 


92850 


12.98651 


94592 


17-49153 


54 


55 


•91140 


10-28610 


92879 


13 •04350 


94621 


17.59139 


55 


56 


•91169 


10.32313 


92908 


13^10096 


94650 


17-69233 


56 


57 


•91197 


10.36040 


92937 


13^15889 


94679 


17.79438 


57 


58 


•91226 


10-39792 


92966 


13.21730 


94708 


17.89755 


58 


59 


•91255 


10-43569 


92995 


13 •27620 


94737 


18-00185 


-^ 


60 


.91284 


10.47371 


93024 


13.33559 


94766 


18.10732 


60 



727 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
87° 88° 89° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 

4 


.94766 
.94795 
.94825 
.94854 
.94883 


18.10732 
18.21397 
18.32182 
18.43088 
18.54119 




.96510 
96539 
96568 
96597 
96626 


27 
27 
28 
28 
28 


65371 
89440 
13917 
38812 
64137 


.98255 
.98284 
.98313 
.98342 
.98371 


53.29869 
57.26976 
58.27431 
59.31411 
60.39105 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.94912 
.94941 
.94970 
.94999 
.95028 


18.65275 
18.76560 
18.87976 
18.99524 
19.11208 




96655 
96684 
96714 
96743 
96772 


28 
29 
29 
29 
29 


89903 
16120 
42802 
69960 
97607 


.98400 
.98429 
.98458 
.98487 
-98517 


61.50715 
62.66460 
63-86572 
65-11304 
66.40927 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.95057 
.95086 
.95115 
.95144 
.95173 


19.23028 
19.34989 
19.47093 
19^59341 
19.71737 




96801 
96830 
96859 
96888 
96917 


30 
30 
30 
31 
31 


25758 
54425 
83623 
13366 
43671 


.98546 
.98575 
.98604 
.98633 
.98662 


67.75736 
69^16047 
70.62285 
72-14583 
73-73586 


10 

11 
12 
13 
14 


15 
16 
17 
18 
11. 


.95202 
.95231 
.95260 
.95289 
.95318 


19.84283 
19.96982 
20-09838 
20-22852 
20-36027 




96946 
96975 
97004 
97033 
97062 


31 
32 
32 
32 
33 


74554 
06030 
38118 
70835 
04199 


.98691 
•98720 
•98749 
.98778 
-98807 


75.39655 
77.13274 
78.94968 
80-85315 
82. 84947 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


.95347 
.95377 
.95406 
.95435 
•95464 


20-49368 
20.62876 
20.76555 
20.90409 
21-04440 




97092 
97121 
97150 
97179 
97208 


33 
33 
34 
34 
34 


38232 
72952 
08380 
44539 
81452 


.98836 
.58866 
-98895 
.98924 
.98953 


84. 94561 
87-14924 
8946886 
91-91387 
94- 49471 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.95493 
.95522 
.95551 
.95580 
.95609 


21.18653 
21.33050 
21.47635 
21.62413 
21. 77386 




97237 
97266 
97295 
97324 
97353 


35 
35 
35 
36 
36 


19141 
57633 
96953 
37127 
78185 


.98982 
.99011 
•99040 
•99069 
•99098 


97-22303 
100.1119 
103.1757 
106.4311 
109-8966 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.95638 
.95667 
.95696 
.95725 
.95754 


21.92559 
22.07935 
22-23520 
22-39316 
22-55328 




97382 
97411 
97440 
97470 
97499 


37 
37 
38 
38 
38 


20155 
63068 
06957 
51855 
97797 


.99127 
•99156 
•99186 
•99215 
-99244 


113-5930 
117-5444 
121.7780 
126.3253 
131-2223 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.95783 
.95812 
.95842 
.95871 
.95900 


22.71563 
22-88022 
23-04712 
23-21637 
23. 38802 




97528 
97557 
97586 
97615 
97644 


39 
39 
40 
40 

41 


44820 
92963 
42266 
92772 
44525 


•99278 
•99302 
•99331 
.99360 
-99389 


136.5111 
142-2406 
148.4684 
155.2623 
162-7033 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


.95929 
.95958 
.95987 
.96016 
.96045 


23-56212 
23-73873 
23-91790 
24-09969 
24-28414 




97673 
97702 
97731 
97760 
97789 


41 
42 
43 
43 
44 


97571 
51961 
07746 
64980 
23720 


•99418 
.99447 
.99476 
.99505 
.99535 


170-8883 
179.9350 
189.9868 
201.2212 
213.8600 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


.96074 
.96103 
.96132 
.96161 
.96190 


24-47134 
24-66132 
24-85417 
25-04994 
25.24869 




97819 
97848 
97877 
97906 
97935 


44 
45 
46 
46 
47 


84026 
45963 
09596 
74997 
42241 


•99564 
•99593 
•99622 
•99651 
•99680 


228.1839 
244.5540 
263^4427 
285^4795 
311.5230 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.96219 
.96248 
.96277 
.96307 
.96336 


25.45051 
25.65546 
25.86360 
26.07503 
26- 28981 




97964 
97993 
98022 
98051 
98080 


48 
48 
49 
50 
51 


11406 
82576 
55840 
31290 
09027 


.99709 
.99738 
.99767 
.99796 
•99825 


342.7752 
380^9723 
428.7187 
490.1070 
571.9581 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


.96365 
.96394 
.96423 
.96452 
.96481 


26.50804 
26.72978 
26.95513 
27.18417 
27-41700 




98109 
98138 
.98168 
.98197 
.98226 


51 
52 
53 
54 
55 


89156 
71790 
57046 
45053 
35946 


.99855 
.99884 
.99913 
.99942 
•99971 


686.5496 
858.4369 
1144.916 
1717.874 
3436.747 


55 
56 
57 
58 
59 


60 


.96510 


27.65371 


.98255 


56 


29869 


1.00000 


Infinite 


60 



728 



TABLE XL— REDUCTION OF BAROxMETER READING TO 32° F. 













Inches. 












Temp. 
























O 

Fahr. 


26-0 


26-5 


27.0 


27.5 


28.0 


28.5 


29.0 


29.5 


300 


30.5 


310 


45 


-039 


-039 


-.040 


-.041 


-.042 


-.042 


-.043 


-.044 


-.045 


-.045 


-.046 


46 


• 041 


.042 


•043 


.043 


•044 


.045 


046 


•046 


.047 


.048 


.049 


47 


■ 043 


• 044 


.045 


•046 


•047 


.048 


.048 


.049 


■ 050 


.051 


.052 


48 


• 046 


• 047 


•047 


• 048 


• 049 


.050 


.051 


.052 


•053 


• 053 


• 054 


49 


• 048 


• 049 


• 050 


.051 


•052 


.052 


.054 


.054 


.055 


.056 


.057 


50 


.050 


• 051 


.052 


.053 


• 054 


• 055 


•056 


• 057 


.058 


.059 


060 


51 


• 053 


• 054 


• 055 


.056 


.057 


.058 


• 059 


.060 


• 061 


• 062 


• 063 


52 


055 


• 056 


057 


•058 


.059 


• 060 


• 061 


•062 


• 064 


.065 


• 066 


53 


• 057 


• 058 


■060 


•061 


.062 


.063 


.064 


.065 


• 066 


• 067 


• 068 


54 


060 


• 061 


•062 


.063 


.064 


• 065 


.067 


.068 


.069 


.070 


.071 


55 


• 062 


• 063 


•064 


.065 


•066 


• 068 


• 069 


.070 


•071 


•073 


.074 


56 


• 064 


■ 065 


• 067 


• 068 


• 069 


.070 


•072 


.073 


• 074 


.075 


■ 077 


57 


• 067 


• 068 


• 069 


• 070 


• 072 


•073 


.075 


• 076 


.077 


.078 


.080 


58 


• 069 


• 070 


■071 


.073 


• 074 


• 076 


•077 


• 078 


.080 


• 081 


.082 


59 


072 


• 073 


.074 


• 075 


• 077 


.078 


.080 


.081 


.083 


.084 


.085 


60 


• 074 


• 076 


• 077 


.078 


• 079 


• 081 


•082 


.084 


.085 


• 086 


.088 


61 


• 076 


• 077 


•079 


• 080 


• 082 


.083 


.085 


.086 


.088 


•089 


.091 


62 


• 079 


• 080 


• 082 


-083 


•085 


.086 


.088 


• 089 


.091 


.092 


• 094 


63 


• 081 


.082 


•084 


.085 


• 087 


088 


.090 


• 091 


.093 


.095 


.096 


64 


• 083 


.085 


.086 


•088 


.090 


.091 


.093 


.094 


.096 


.097 


•099 


65 


• 086 


• 087 


.089 


•090 


• 092 


.093 


.095 


• 097 


•099 


.100 


.102 


66 


• 088 


• 089 


• 091 


.093 


.095 


.096 


.098 


.099 


.101 


.103 


• 105 


67 


• 090 


• 092 


•094 


• 095 


097 


.099 


.101 


.102 


• 104 


.106 


.108 


68 


•093 


.094 


.096 


•098 


.100 


.101 


.103 


.105 


.107 


.108 


• 110 


69 


•095 


• 097 


.099 


• 100 


.102 


.104 


.106 


.107 


.110 


.111 


.113 


70 


.097 


.099 


.101 


• 103 


.105 


.106 


.109 


.110 


.112 


.114 


.116 


71 


.100 


.101 


.103 


.105 


.107 


.109 


.111 


.113 


.115 


.117 


.119 


72 


.102 


.104 


.106 


.108 


.110 


.112 


• 114 


• 116 


.118 


.120 


.122 


73 


• 104 


.106 


.108 


.110 


.112 


.114 


.116 


.118 


.120 


.122 


.124 


74 


• 107 


.109 


.111 


.113 


.115 


.117 


.119 


.121 


.123 


.125 


.127 


75 


.109 


• 111 


.113 


.115 


.117 


.119 


.122 


.124 


.126 


.128 


.130 


76 


• 111 


.113 


.116 


• 118 


.120 


• 122 


.124 


.126 


.128 


• 130 


.133 


77 


• 114 


.116 


.118 


• 120 


.122 


.124 


.127 


• 129 


.131 


.133 


.136 


78 


• 116 


.118 


.120 


.122 


.125 


.127 


.129 


• 131 


.134 


.136 


.138 


79 


.118 


.120 


.123 


.125 


.127 


.129 


.132 


.134 


.137 


• 139 


.141 


80 


• 121 


.123 


.125 


.127 


.130 


.132 


.135 


• 137 


.139 


.141 


,.144 


81 


• 123 


.125 


.128 


.130 


.132 


.134 


.137 


.139 


• 142 


.144 


.147 


82 


• 125 


.128 


.130 


.132 


.135 


.137 


.140 


.142 


.145 


• 147 


.149 


83 


• 128 


.130 


.133 


• 135 


.138 


.140 


.142 


.145 


• 147 


.149 


.152 


84 


.130 


.132 


• 135 


■ 138 


.140 


.142 


.145 


.147 


.150 


• 152 


.155 


85 


• 132 


• 134 


.137 


.140 


.143 


.145 


• 148 


• 150 


.153 


.155 


.158 


86 


.135 


.137 


.140 


.142 


.145 


• 148 


• 150 


.153 


.155 


• 158 


.161 


87 


• 137 


.139 


.142 


.144 


.148 


• 150 


.153 


.155 


.158 


.161 


.163 


88 


.139 


.142 


• 145 


.147 


.150 


• 152 


.155 


.158 


.161 


.163 


.166 


89 


• 142 


• 144 


• 147 


.150 


• 153 


.155 


.158 


.161 


.164 


.166 


.169 


90 


• 144 


.147 


.150 


.153 


.155 


.158 


.161 


.164 


.166 


.169 


.172 


91 


-.146 


-.149 


-.152 


-.155 


-.158 


-.160 


-.163 


-.166 


-.169 


-.172 


-.175 



729 



TABLE XII.— BAROMETRIC ELEVATIONS.* 



Inches. 



20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
23 
23 
23 
23 
23 
23 
23 
23. 



Feet. 

11,047 
10,911 
10,776 
10,642 
10,508 
10.375 
10,242 
10,110 
9.979 
9,848 
9,718 
9.589 
9.460 
9.332 
9-204 
9.077 
8,951 
8.825 
8. 700 
8.575 
8.451 
8.327 
8.204 
8.082 
7,960 
7,838 
7.717 
7.597 
7.477 
7358 
7.239 
7.121 
7.004 
6,887 
6,770 
6,554 
6,538 
6,423 



Diff. for 
• 01. 



Feet. 



13 


6 


13 


5 


13 


4 


13 


4 


13 


3 


13 


3 


13 


2 


13 


1 


13 


1 


13 





12 


9 


12 


9 


12 


8 


12 


8 


12 


7 


12 


6 


12 


6 


12 


5 


12 


5 


12 


4 


12 


4 


12 


3 


12 


2 


12 


2 


12 


2 


12 


1 


12 





12 







9 




9 




8 




7 




7 




7 




6 




6 




5 



B 




A 


Inches. 


Feet. 


23-7 


6.423 


23 


8 


6,308 


23 


9 


6.194 


24 





6,080 


24 


1 


5.967 


24 


2 


5854 


24 


3 


5.741 


24 


4 


5,629 


24 


5 


5,518 


! 24 


6 


5-407 


; 24 


7 


5.296 


24 


8 


5.186 


24 


9 


5077 


i 25 





4-968 


i 25 


1 


4.859 


1 25 


2 


4.751 


i 25 


3 


4.643 


1 25 


4 


4 535 


' 25 


5 


4-428 


; 25 


6 


4-321 


25 


7 


4-215 


25 


8 


4-109 


25 


9 


4,004 


26 





3 899 


26 


1 


3.794 


26 


2 


3 690 


26 


3 


3. 586 


26 


4 


3-483 


26 


5 


3.380 


26 


6 


3 277 


26 


7 


3.175 


. 26 


8 


3.073 


1 26 


9 


2 972 


27 





2871 


27 


1 


2 770 


27 


2 


2. 670 


27 


3 


2,570 


27 


4 


2 470 



Diff. 


for . 


• 01. 


Feet. 


-11.5 


11 


4 


11 


4 


11 


3 


11 


3 


11 


3 


11 


2 


11 


1 


11 


1 


11 


1 i 


11 


i 


10 


9 1 


10 


9 


10 


9 ! 


10 


8 i 


10 


8 


10 


8 


10 


7 


10 


7 ! 


10 


6 I 


10 


6 i 


10 


5 ' 


10 


5 


10 


4 


10 


4 


10 


4 


10 


3 


10 


3 


10 


3 i^ 


10 


2 1 


10 


2 1 


10 


1 ' 


10 


1 


10 


1 


10 





10 





-10 






Inches. 



27 
27 
27 
27 
27 
27 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
31 



Feet. 

2,470 

2,371 

2,272 

2,173 

2-075 

1.977 

1 880 

1.783 

] 686 

1,589 

1.493 

1.397 

1302 

1-207 

1.112 

1.018 

924 

830 

736 

643 

550 

458 

366 

274 

182 

91 



-91 

181 

271 

361 

451 

540 

629 

717 

805 

-893 



Diff. for 
01- 



* Compiled from Report of U. S. C. & G. Survey for 1881., App. 10 Table XL 



TABLE XIII.- 



-COEFFICIENTS FOR CORRECTIONS FOR TEMPERATURE 
AND HUMIDITY.* 



t-\-t' 

0° 
10 
20 
30 
40 
50 
60 



■•1024 
.0915 
.0806 

• 0698 

• 0592 
.0486 

■0380 



Diff. for 
1°. 



t+v 


c 


60° - 


0380 


70 


0273 


80 


0166 


90 - 


0058 


100 + 


0049 


110 


0156 


120 + 


0262 



Diff. for 

1°. 


t + t' 


C 




120° + 


0262 


10-7 


130 


0368 


10-7 1 


140 


0472 


108 


150 


0575 


10-7 


160 


0677 


10-7 


170 


0779 


10-6 


180 + 


0879 



* Compiled from Report of U. S. C. & G. Survey for 1881, App. 10, Tables I, IV. 

7.S0 



TABLE XXX.— USEFUL TRIGONOMETMCAL FORMULA-:. 



10 



tan a 



_ __1 tana _ / I — c os 2a 1 

~coseca~Vl + tan2 a" V ^ "~Vi + cot2 a 

= cos a tan a = V 1 — cos- a = 2 sin ^a cos ^a 
_ 1 + cos a _ 2 tan ^a 



cot ^a 1 + tan2 ^a 



= vers a cot ^a. 



cos a =- 



1 



cot a 



sec a Vi + cot2 a Vl+tan2o 



= 1— vers a=sin a cot a = Vl— sin^ a = 2 cos^ ^a — 1 
= sin a cot ^a — 1 = cos^ ^a — sin^ ^a=l — 2 sin^ ^a. 



1 



sin a sec a 



1 



cot a cos a cosec a v^cosec^ a — 1 
= vers 2a cosec 2a = cot a — 2 cot 2a = sin a sec a 
sin 2a 



cot a = 



1 + cos 2a 
1 c 



= exsec a cot ^a = oxsec 2a cot 2a. 
OS a sin 2a 1 + cos 2a 



tan a sin a 1 — cos 2a sin 2a 
= V cosec2 a — 1 = cot ^a — cosec a. 
vers a = 1 — cos a -=sin a tan ha = 2 sin^ ^a = cos a exsec a, 
exsec a = sec a — 1 = tan a tan +a =- vers a sec a. 



. 1 /vers a 

sin ia = i/ — y- = 



sin a _ vers a cos ^a 
2 cos \a sin a 



cos ^a 



/ 1 + cos a _ sin a _ sin a sin ^a 
f 2 2 sin ^a verso ' 



tan ^a = vers a cosec a = cosec a — cot a 



tan a 



„ . 1 1+cos a 

cot ^a = : = cosec a + cot a = 



1 +sec a 
tan a 



1 



exsec a cosec a — cot a ' 



vers ia = 1 - V^(i + cos a). 
1 



exsec ^a = - 



VA(l + cosa) 



-1. 



731 



TABLE XXX.— USEFUL TRIGONOMETRICAL FORMULA. 



sin 2a 



^ 2 ttiu a cos a — 



2 tan a 



1 + tan2 a" 

cos 2a = cos2 a — sin^ a = 1 — 2 sin^ a = 2 cos^ a — I 
1 - taii2 a 



tan 2a 



1 + tan2 a* 



2 tan g 
1 — tan- a' 



cot 2a =i cot g-i tan a = ^^^^ ^- ^ ^ ^ " tan^ g^ 
2 cot a 2 tan a 

vers 2a = 2 sin^ a - 1 - cos 2g = 2 sin a cos a tan a. 

exsec 2a = i^^B^ ^Jjari^ g ^ 2 sin^^ g 

cot a 1 — tan- g 1—2 sin- a ' 

sin (g ± 6) =sin g cos b ± cos g sin b. 

cos (g ± ^) = cos g cos 6 T sin a sin 6. 

sin g +sin 6=2 sin K'^ + 6) cos ^(g - 6;. 

sin g — sin 6^2 sin ^(g — 6) oos ^(a + b). 

cos g + cos 6 = 2 cos i(g-l-6) cos ^(g — 6). 

cos a — cos 6 = — 2 sin ^(a -f- 6) sin ^ (a — 6). 



Call the sides of any triangle A, B, C, and the opposite angles a, h 
and c. Calls = ^(A+^-|-C'). 

tan^(g — 6) =^-r—~~ta.n ^(a~\-b) = \ , .. cot ^c. 
A -r -tf A + £> 

C = U +B)^^4^ =(.4 -B)5iiii^> 

c()S^(g-6) ^sin^(a-6) 



sinig = ^. 



/(s-B)is-C) 



2(s-B)(s-C) 
versg = ^ . 



Area =Vs(s-A)(s-B)(s-C)^A''~ 



sin 6 sin c 



2 sin a 



732 



TABLE XXXI.— USEFUL FORMULA. AND CONSTANTS. 



Circuaiference of a circle (radius = r) = 2nr. 

Area of a circle = nr^. 

Area of sector (length of arc = /) = lh\ 

a 

■' " " (angle of arc = a°) = qah^*'^- 

Area of segment (chord = c, mid. ord. = m) = fern (approx.). 

Area of a circle to radius 1 1 

Circumference of a circle to diameter 1 )■ = n =. 3.1415927 

I 
Surface of a sphere to diameter 1 J 

Volume of a sphere to radius 1 = 47r -^ 3 = 4.1887902 

r degrees = 57.2957795 

I 
Arc equal to radius expressed in ■{ minutes = 3437.7467708 

I 
L seconds = 206264.8062471 

Length of arc of 1°, radius unity 0.01745329 

Sine of one second = 0.0000048481 

Cubic inches in United States standard gallon = 231 

Weight of one cubic foot of water at maximum density (therm. 

39°.8 F., barom. 30^0 62.379 

Weight of one cubic foot of water at ordinary temperature (therm. 

62° F.) 62.321 

Acceleration due to gravity at latitude of New York in feet per 

square second 32.15945 

Feet in one metre 3.280869 

Metres in one foot 0.304797 

733 



Logarithm. 



0.4971499 

. 622 0886 
1.7581226 
3.5.36 2739 
5.314 4251 
8.2418774 
4.6855749 
2.363 6120 

1.795 0384 

1 . 794 6349 

1.507 3086 
0.515 9889 
9.4840111 



TABT.E XXXII— SQUARES, CUBES, SQITARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


1 


1 


1 


1.0000000 


1.0000000 


1.000000000 


2 


4 


8 


1.4142136 


1.2599210 


.500000000 


3 


9 


27 


1.7320508 


1.4422496 


.333333333 


4 


16 


64 


2.0000000 


1.5874011 


.250000000 


5 


25 


125 


2.2360680 


1.7099759 


- 200000000 


6 


36 


216 


2.4494897 


1.8171206 


.166666667 


7 


49 


343 


2.6457513 


1.9129312 


.142857143 


8 


64 


512 


2.8284271 


2.0000000 


.125000000 


9 


81 


729 


3.0000000 


2.0800837 


.111111111 


10 


100 


1000 


3.1622777 


2.1544347 


.100000000 


11 


121 


1331 


3.3166248 


2.2239801 


-090909091 


12 


144 


1728 


3.4641016 


2-2894286 


.083333333 


13 


169 


2197 


3.6055513 


2.3513347 


.076923077 


14 


196 


2744 


3-7416574 


2.4101422 


-071428571 


15 


225 


3375 


3-8729833 


2-4662121 


066666667 


16 


256 


4096 


4.0000000 


2.5198421 


-062500000 


17 


289 


4913 


4-1231056 


2-5712816 


.058823529 


18 


324 


5832 


4-2426407 


2-6207414 


.055555556 


19 


361 


6859 


4-3588989 


2-6684016 


-052631579 


•>o 


400 


8000 


4-4721360 


2-7144177 


-050000000 


21 


441 


9261 


4-5825757 


2-7589243 


-047619048 


22 


484 


10648 


4-6904158 


2-8020393 


.045454545 


23 


529 


12167 


4-7958315 


2-8438670 


.043478261 


24 


576 


13824 


4-8989795 


2.8844991 


.041666667 


25 


625 


15625 


5-0000000 


2-9240177 


-040000000 


26 


676 


17576 


5-0990195 


2.9624960 


.038461538 


27 


729 


19683 


5-1961524 


3.0000000 


-037037037 


28 


784 


21952 


5.2915026 


3-0365889 


.035714286 


29 


841 


24389 


5.3851648 


3.0723168 


.034482759 


30 


900 


27000 


5-4772256 


3-1072325 


-033333333 


31 


961 


29791 


5.5677644 


3.1413806 


-032258065 


32 


1024 


32768 


5.6568542 


3.1748021 


-031250000 


33 


1089 


35937 


5.7445626 


3.2075343 


-030303030 


34 


1156 


39304 


5.8309519 


3.2396118 


.029411765 


35 


1225 


42875 


5-9160798 


3-2710663 


.028571429 


36 


1296 


46656 


6.0000000 


3.3019272 


.027777778 


37 


1369 


50653 


6-0827625 


3.3322218 


.027027027 


38 


1444 


54872 


6-1644140 


3-3619754 


.026315789 


39 


1521 


59319 


6-2449980 


3.3912114 


.025641026 


40 


1600 


64000 


6-3245553 


3-4199519 


-025000000 


41 


1681 


68921 


6.4031242 


3-4482172 


-024390244 


42 


1764 


74088 


6.4807407 


3-4760266 


.023809524 


43 


1849 


79507 


6.5574385 


3-5033981 


.023255814 


44 


1936 


85184 


6.6332496 


3.5303483 


.022727273 


45 


2025 


91125 


6.7082039 


3-5568933 


-022222222 


46 


2116 


97336 


6.7823300 


3-5830479 


.021739130 


47 


2209 


103823 


6.8556546 


3.6088261 


.021276600 


48 


2304 


110592 


6-9282032 


3.6342411 


.020833333 


49 


2401 


117649 


7-0000000 


3.6593057 


.020408163 


50 


2500 


125000 


7.0710678 


3.6840314 


-020000000 


51 


2601 


132651 


7.1414284 


3.7084298 


.019607843 


52 


2704 


140608 


7.2111026 


3.7325111 


.019230769 


53 


2809 


148877 


7.2801099 


3.7562858 


.018867925 


54 


2916 


157464 


7.3484692 


3.7797631 


.018518519 


55 


3025 


166375 


7-4161985 


3.8029525 


.018181818 


56 


3136 


175616 


7.4833148 


3.8258624 


.017857143 


57 


3249 


185193 


7.5498344 


3.8485011 


.017543860 


58 


3364 


195112 


7.6157731 


3.8708766 


.017241379 


59 


3481 


205379 


7.6811457 


3.8929965 


.016949153 


60 


3600 


216000 


7.7459667 


3.9148676 


.016666667 



734 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. E 


reciprocals. 


61 


3721 


226981 


7.8102497 


3-9364972 


016393443 


62 


3844 


238328 


7-8740079 


3-9578915 


016129032 


63 


3969 


250047 


7-9372539 


3-9790571 


015873016 


64 


4096 


262144 


8-0000000 


4-0000000 


015625000 


65 


4225 


274625 


8-0622577 


4-0207256 


015384615 


66 


4356 


287496 


8-1240384 


4-0412401 


015151515 


67 


4489 


300763 


8-1853528 


4-0615480 


014925373 


68 


4624 


314432 


8-2462113 


4.0816551 


014705882 


69 


4761 


328509 


8-3066239 


4-1015661 


014492754 


70 


4900 


343000 


8-3666003 


4-1212853 


014285714 


71 


5041 


357911 


8-4261498 


4-1408178 


014084507 


72 


5184 


373248 


8-4852814 


4-1601676 


013888889 


73 


5329 


389017 


8-5440037 


4.1793390 


013698630 


74 


5476 


405224 


8-6023253 


4.1983364 


013513514 


75 


5625 


421875 


8-6602540 


4-2171633 


013333333 


76 


5776 


438976 


8-7177979 


4-2358236 


013157895 


77 


5929 


456533 


8-7749644 


4-2543210 ■ . 


012987013 


78 


6084 


474552 


8-8317609 


4-2726586 


012820513 


79 


6241 


493039 


8-8881944 


4-2908404 


012658228 


80 


6400 


512000 


8-9442719 


43088695 


012500000 


81 


6561 


531441 


9-0000000 


4-3267487 


012345679 


82 


6724 


551368 


9-0553851 


4-3444815 


012195122 


83 


6889 


571787 


9-1104336 


4.3620707 


012048193 


84 


7056 


592704 


9-1651514 


4-3795191 


011904762 


85 


7225 


614125 


9-2195445 


4-3968296 


011764706 


86 


7396 


636056 


9.2736185 


4-4140049 


011627907 


87 


7569 


658503 


9-3273791 


4-4310476 


011494253 


88 


7744 


681472 


9.3808315 


4-4479602 


011363636 


89 


7921 


704969 


9-4339811 


4-4647451 


011235955 


90 


8100 


729000 


9- 4868330 


4-4814047 


011111111 


91 


8281 


753571 


9-5393920 


4-4979414 


010989011 


92 


8464 


778688 


9-5916630 


4.5143574 


010869565 


93 


8649 


804357 


9.6436508 


4.5306549 


010752688 


94 


8836 


830584 


9-6953597 


4.5468359 


010638298 


95 


9025 


857375 


9-7467943 


4-5629026 


010526316 


96 


9216 


884736 


9.7979590 


4-5788570 


010416667 


97 


9409 


912673 


9.8488578 


4.5947009 


010309278 


98 


9604 


941192 


9-8994949 


4-6104363 


010204082 


99 


9801 


970299 


9-9498744 


4-6260650 


010101010 


100 


10000 


1000000 


10-0000000 


4-6415888 


010000000 


101 


10201 


1030301 


10.0498756 


4-6570095 


009900990 


102 


10404 


1061208 


10.0995049 


4-6723287 


009803922 


103 


10609 


1092727 


10.1488916 


4-6875482 


009708738 


104 


10816 


1124864 


10.1980390 


4-7026694 


009615385 


105 


11025 


1157625 


10-2469508 


4-7176940 


009523810 


106 


11236 


1191016 


10.2956301 


4.7326235 


009433962 


107 


11449 


1225043 


10.3440804 


4-7474594 


009345794 


108 


11664 


1259712 


10-3923048 


4-7622032 


009259259 


109 


11881 


1295029 


10.4403065 


4-7768562 


009174312 


110 


12100 


1331000 


10-4880885 


4-7914199 


009090909 


111 


12321 


1367631 


10.5356538 


4.8058955 


.009009009 


112 


12544 


1404928 


10.5830052 


4.8202845 


.008928571 


113 


12769 


1442897 


10-6301458 


4.8345881 


.008849558 


114 


12996 


1481544 


10-6770783 


4.8488076 


.008771930 


115 


13225 


1520875 


10.7238053 


4-8629442 


-008695652 


116 


13456 


1560896 


10-7703296 


4.8769990 


.008620690 


117 


13689 


1601613 


10-8166538 


4.8909732 


.008547009 


118 


13924 


1643032 


10-8627805 


4.9048681 


.008474576 


119 


14161 


1685159 


10-9087121 


4.9186847 


.008403361 


120 


14400 


1728000 


10-9544512 


4.9324242 


.008333333 



735 





TABLE XXXII.— SQUARES 


, CUBES, 


SQUARE ROOTS, 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


121 
122 
123 
124 
125 


14641 
14884 
15129 
15376 
15625 


1771561 
1815848 
1860867 
1906624 
1953125 


11. 

11. 

11 

11 

11 


0000000 
0453610 
0905365 
1355287 
1803399 


4 
4 
4 
4 
5 


9460874 
9596757 
9731898 
9866310 
0000000 




008264463 
008196721 
008130081 
008064516 
008000000 


126 
127 
128 
129 
130 


15876 

16129 
16384 
16641 
16900 


2000378 
2048383 
2097152 
2146689 
2197000 


11 
11 
11 
11 
11 


2249722 
2694277 
3137085 
3578167 
4017543 


5 
5 
5 
5 
5 


0132979 
0265257 
0396842 
0527743 
0657970 




007936508 
007874016 
007812500 
007751938 
007692308 


131 
132 
133 
134 
135 


17161 
17424 
17689 
17956 
18225 


2248091 
2299968 
2352637 
2406104 
2460375 


11 
11 
11 
11 
11 


4455231 
4891253 
5325626 
5758369 
6189500 


5 
5 
5 
5 
5 


0787531 
0916434 
1044687 
1172299 
1299278 




007633588 , 

007575758 

007518797 

007462687 

007407407 


136 
137 
138 
139 
140 


18496 
18769 
19044 
19321 
19600 


2515456 
2571353 
2628072 
2685619 
2744000 


11 
11 
11 
11 
11 


6619038 
7046999 
7473401 
7898261 
8321596 


5 
5 
5 
5 
5 


1425632 
1551367 
1676493 
1801015 
1924941 




007352941 
007299270 
007246377 
007194245 
007142857 


141 
142 
143 
144 
145 


19881 
20164 
20449 
20736 
21025 


2803221 
2863288 
2924207 
2985984 
3048625 


11 
11 
11 
12 
12 


8743421 
9163753 
9582607 
0000000 
0415946 


5 
5 
5 
5 
5 


2048279 
2171034 
2293215 
2414828 
2535879 




007092199 
007042254 
006993007 
006944444 
006896552 


146 
147 
148 
149 
150 


21316 
21609 
21904 
22201 
22500 


3112136 
3176523 
3241792 
3307949 
3375000 


12 
12 
12 
12 
12 


0830460 
1243557 
1655251 
2065556 
2474487 


5 
5 
5 
5 
5 


2656374 
2776321 
2895725 
3014592 
3132928 




006849315 
006802721 
006756757 
006711409 
006666667 


151 
152 
153 
154 
155 


22801 
23104 
23409 
23716 
24025 


3442951 
3511808 
3581577 
3652264 
3723875 


12 
12 
12 
12 
12 


2882057 
3288280 
3693169 
4096736 
4498996 


5 
5 
5 
5 
5 


3250740 
3368033 
3484812 
3601084 
3716854 




•006622517 
006578947 
006535948 
006493506 
006451613 


156 
157 
158 
159 
160 


24336 
24649 
24964 
25281 
25600 


3796416 
3869893 

3944312 
4019679 
4096000 


12 
12 
12 
12 
12 


4899960 
5299641 
5698051 
6095202 
6491106 


5 
5 
5 
5 
5 


3832126 
3946907 
4061202 
4175015 
4288352 




006410256 
.006369427 
•006329114 
■006289308 
•006250000 


161 
162 
163 
164 
165 


25921 
26244 
26569 
26896 
27225 


4173281 
4251528 
4330747 
4410944 
4492125 


12 
12 
12 
12 
12 


6885775 
7279221 
7671453 
8062485 
8452326 


5 
5 
5 
5 
5 


4401218 
4513618 
4625556 
4737037 
4848066 




•006211180 
•006172840 
■006134969 
•006097561 
•006060606 


166 
167 
168 
169 
170 


27556 
27889 
28224 
28561 
28900 


4574296 
4657463 
4741632 
4826809 
4913000 


12 
12 
12 
13 
13 


8840987 
9228480 
9614814 
0000000 
0384048 


5 
5 
5 
5 
5 


4958647 
5068784 
•5178484 
5287748 
5396583 




•006024096 
•005988024 
•005952381 
■005917160 
•005882353 


171 
172 
173 
174 
175 


29241 
29584 
29929 
30276 
30625 


5000211 
5088448 
5177717 
5268024 
5359375 


13 
13 
13 
13 
13 


0766968 
1148770 
1529464 
1909060 
■2287566 


5 
5 
5 
5 
5 


■5504991 
5612978 
•5720546 
.5827702 
.5934447 




.005847953 
.005813953 
.005780347 
•005747126 
005714286 


176 
177 
178 
179 
180 


30976 
31329 
31684 
32041 
32400 


5451776 
5545233 
5639752 
5735339 
5832000 


13 
13 
13 
13 
13 


•2664992 
•3041347 
•3416641 
•3790882 
.4164079 


5 
5 
5 
5 
5 


6040787 
6146724 
6252263 
6357408 
6462162 




.005681818 
005649718 
005617978 
005586592 
005555556 



736 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


181 
182 
183 
184 
185 


32761 
33124 
33489 
33856 
34225 


5929741 
6028568 
6128487 
6229504 
6331625 


13 
13 
13 
13 
13 


•4536240 
.4907376 
•5277493 
•5646600 
•6014705 


5 
5 
5 
5 
5 


•6566528 
•6670511 
•6774114 
• 6877340 
•6980192 




•005524862 
•005494505 
•005464481 
•005434783 
•005405405 


186 
187 
188 
189 
190 


34596 
34969 
35344 
35721 
36100 


6434856 
6539203 
6644672 
6751269 
6859000 


13 
13 
13 
13 
13 


•6381817 
6747943 
•7113092 
•7477271 
7840488 


5 
5 
5 
5 
5 


•7082675 
•7184791 
•7286543 
•7387936 
•7488971 




.005376344 
-005347594 
.005319149 
.005291005 
•005263158 


191 
192 
193 
194 
195 


36481 
36864 
37249 
37636 
38025 


69-67871 
7077888 
7189057 
7301384 
7414875 


13 
13 
13 
13 
13 


8202750 
•8564065 
•8924440 
9283883 
9642400 


5 
5 
5 
5 
5 


•7589652 
■7689982 
•7789966 
•7889604 
•7988900 




.005235602 
•005208333 
.005181347 
.005154639 
.005128205 


196 
197 
198 
199 
*^00 


38416 
38809 
39204 
39601 
40000 


7529536 
7645373 
7762392 
7880599 
8000000 


14 
14 
14 
14 
14 


•0000000 
0356688 

•0712473 
1067360 
1421356 


5 
5 
5 
5 
5 


•8087857 
8186479 
■8284767 
■8382725 
.8480355 




.005102041 
.005076142 
.005050505 
005025126 
005000000 


201 
202 
203 
204 
205 


40401 
40804 
41209 
41616 
42025 


8120601 
8242408 
8365427 
8489664 
8615125 


14 
14 
14 
14 
14 


1774469 
•2126704 
•2478068 

2828569 
.3178211 


5 
5 
5 
5 
5 


8577660 
8674643 
8771307 
8867653 
8963685 




.004975124 
004950495 
004926108 
004901961 
004878049 


206 
207 
208 
209 
310 


42436 
42849 
43264 
43681 
44100 


8741816 
8869743 
8998912 
9129329 
9261000 


14 
14 
14 
14 
14 


•3527001 
•3874946 
•4222051 
•4568323 
•4913767 


5 
5 
5 
5 
5 


9059406 
9154817 
9249921 
9344721 
9439220 




004854369 
004830918 
004807692 
004784689 
004761905 


211 
212 
213 
214 
215 


44521 
44944 
45369 
45796 
46225 


9393931 
9528128 
9663597 
9800344 
9938375 


14 
14 
14 
14 
14 


.5258390 
•5602198 
•5945195 
•6287388 
6628783 


5 
5 
5 
5 
5 


9533418 
9627320 
9720926 
9814240 
9907264 




004739336 
004716981 
004694836 
004672897 
004651163 


216 
217 
218 
219 
220 


46656 
47089 
47524 
47961 
48400 


10077696 
10218313 
10360232 
10503459 
10648000 


14 
14 
14 
14 
14 


6969385 
7309199 
7648231 
7986486 
8323970 


6 
6 
6 
6 
R 


0000000 
0092450 
0184617 
0276502 
0368107 




004629630 
004608295 
004587156 
004566210 
004545455 


221 
222 
223 
224 
225 


48841 
49284 
49729 
50176 
50625 


10793861 
10941048 
11089567 
11239424 
11390625 


14 
14 
14 
14 
15 


8660687 
8996644 
9331845 
9666295 
0000000 


6 
6 
6 
6 
6 


0459435 
0550489 
0641270 
0731779 
0822020 




004524887 
004504505 
004484305 
004464286 
004444444 


226 
227 
228 
229 
230 


51076 
51529 
51984 
52441 
52900 


11543176 
11697083 
11852352 
12008989 
12167000 


15 
15 
15 
15 
15 


0332964 
0665192 
0e96689 
1327460 
1657509 


6 
6 
6 
6 
6 


0911994 
1001702 
1091147 
1180332 
1269257 




004424779 
004405286 
004385965 
004366812 
004347826 


231 
232 
233 
234 
235 


53361 
53824 
54289 
54756 
55225 


12326391 
12487168 
12649337 
12812904 
12977875 


15 
15 
15 
15 
15 


1986842 
2315462 
2643375 
2970585 
3297097 


6 
6 
6 
6 
6 


1357924 
1446337 
1534495 
1622401 
1710058 




004329004 
004310345 
004291845 
004273504 
004255319 


236 
237 
238 
239 
240 


55696 
56169 
56644 
57121 
57600 


13144256 
13312053 
13481272 
13651919 
13824000 


15 
15 
15 
15. 
15. 


3622915 
3948043 
4272486 
4596248 
4919334 


6 

6 

6. 

6. 

6. 


1797466 
1884628 
1971544 
2058218 
2144650 




004237288 
004219409 
004201681 
004184100 
004166667 



737 





TABLE XXXII.— SQUARES 


, CUBES, 


SQUARE ROOTS, 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


241 
242 
243 
244 
245 


58081 
58564 
59049 
59536 
60025 


13997521 
14172488 
14348907 
14526784 
14706125 


15. 
15. 
15. 
15. 
15. 


5241747 
5563492 
5884573 
6204994 
6524758 


6. 
6. 
6. 
6. 
6. 


2230843 
2316797 
2402515 
2487998 
2573248 




004149378 
004132231 
004115226 
004098361 
004081633 


246 
247 
248 
249 
250 


60516 
61009 
61504 
62001 
62500 


14886936 
15069223 
15252992 
15438249 
15625000 


15. 
15. 
15. 
15. 
15. 


6843871 
7162336 
7480157 
7797338 
8113883 


6 
6 
6 
6 
6 


2658266 
2743054 
2827613 
2911946 
2996053 




004065041 
004048583 
004032258 
004016064 
004000000 


251 
252 
253 
254 
255 


63001 
63504 
64009 
64516 
65025 


15813251 
16003008 
16194277 
16387064 
16581375 


15. 
15. 
15. 
15 
15 


8429795 
8745079 
9059737 
9373775 
9687194 


6 
6 
6 
6 
6 


3079935 
3163596 
3247035 
3330256 
3413257 




003984064 
003968254 
003952569 
003937008 
003921569 


256 
257 
258 
259 
260 


65536 
66049 
66564 
67081 
67600 


16777216 
16974593 
17173512 
17373979 
17576000 


16 
16 
16 
16 
16 


0000000 
0312195 
0623784 
0934769 
1245155 


6 
6 
6 
6 
6 


3496042 
3578611 
3660968 
3743111 
3825043 




003906250 
003891051 
003875969 
003861004 
003846154 


261 
262 
263 
264 
265 


68121 
68644 
69169 
69696 
70225 


17779581 
17984728 
18191447 
18399744 
18609625 


16 
16 
16 
16 
16 


1554944 
1864141 
2172747 
2480768 
2788206 


6 
6 
6 
6 
6 


3906765 
3988279 
4069585 
4150687 
4231583 




003831418 
003816794 
003802281 
003787879 
003773585 


266 
267 
268 
269 
270 


70756 
71289 
71824 
72361 
72900 


18821096 
19034163 
19248832 
19465109 
19683000 


16 
16 
16 
16 
16 


3095064 
3401346 
3707055 
4012195 
4316767 


6 
6 
6 
6 
6 


4312276 
4392767 
4473057 
4553148 
4633041 




003759398 
003745318 
003731343 
003717472 
003703704 


271 
272 
273 
274 
275 


73441 
73984 
74529 
75076 
75625 


19902511 
20123648 
20346417 
20570824 
20796875 


16 
16 
16 
16 
16 


4620776 
4924225 
5227116 
5529454 
5831240 


6 
6 
6 
6 
6 


4712736 
4792236 
4871541 
4950653 
5029572 




.003690037 
.003676471 
■003663004 
■003649635 
•003636364 


276 
277 
278 
279 
280 


76176 
76729 
77284 
77841 
78400 


21024576 
21253933 
21484952 
21717639 
21952000 


16 
16 
16 
16 
16 


6132477 
6433170 
6733320 
7032931 
7332005 


6 
6 
6 
6 
6 


5108300 
5186839 
5265189 
•5343351 
•5421326 




•003623188 
•003610108 
•003597122 
•003584229 
•003571429 


281 
282 
283 
284 
285 


78961 
79524 
80089 
80656 
81225 


22188041 
22425768 
22665187 
22906304 
23149125 


16 
16 
16 
16 
16 


7630546 
7928556 
8226038 
8522995 
8819430 


6 
6 
6 
6 
6 


•5499116 
•5576722 
•5654144 
•5731385 
•5808443 




.003558719 
003546099 
•003533569 
•003521127 
•003508772 


286 
287 
288 
289 
290 


81796 
82369 
82944 
83521 
84100 


23393656 
23639903 
23887872 
24137569 
24389000 


16 
16 
16 
17 
17 


9115345 
9410743 
9705627 
0000000 
0293864 


6 
6 
6 
6 
6 


•5885323 
•5962023 
6038545 
•6114890 
•6191060 




•003496503 
.003484321 
•003472222 
.003460208 
•003448276 


291 
292 
293 
294 
295 


84681 
85264 
85849 
86436 
87025 


24642171 
24897088 
25153757 
25412184 
25672375 


17 
17 
17 
17 
17 


■0587221 
0880075 
■1172428 
.1464282 
.1755640 


6 
6 
6 
6 
6 


6267054 

•6342874 

•6418522 

6493998 

6569302 




•003436426 
•003424658 
•003412969 
.003401361 
•003389831 


296 
297 
298 
299 
300 


87616 
88209 
88804 
89401 
90000 


25934336 
26198073 
26463592 
26730899 
27000000 


17 
17 
17 
17 
17 


2046505 
.2336879 
•2626765 
.2916165 
.3205081 


6 
6 
6 
6 
6 


•6644437 
6719403 
6794200 
■6868831 
■6943295 




.003378378 
003367003 
.003355705 
.003344482 
•003333333 



738 



CUBE ROOTS, AND RECIPIIOCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


301 
302 
303 
304 
305 


90601 
91204 
91809 
92416 
93025 


27270901 
27543608 
27818127 
28094464 
28372625 


17.3493516 
17.3781472 
17.4068952 
17.4355958 
17.4642492 


6 
6 
6 
6 
6 


7017593 
7091729 
7165700 
7239508 
7313155 




003322259 
003311258 
003300330 
003289474 
003278689 


306 
307 
308 
309 
310 


93636 
94249 
94864 
95481 
96100 


28652616 
28934443 
29218112 
29503629 
29791000 


17.4928557 
17.5214155 
17.5499288 
17.5783958 
17.6068169 


6 
6 
6 
6 
6 


7386641 
7459967 
7533134 
7606143 
7678995 




003267974 
003257329 
003246753 
003236246 
003225806 


311 
312 
313 
314 
315 


96721 
97344 
97969 
98596 
99225 


30080231 
30371328 
30664297 
30959144 
31255875 


17.6351921 
17.6635217 
17.6918060 
17.7200451 
17.7482393 


6 
6 
6 
6 
6 


7751690 
7824229 
7896613 
7968844 
8040921 




003215434 
003205128 
003194888 
003184713 
003174603 


316 
317 
318 
319 
320 


99856 
100489 
101124 
101761 
102400 


31554496 
31855013 
32157432 
32461759 
32768000 


17.7763888 
17.8044938 
17.8325545 
17.8605711 
17.8885438 


6 
6 
6 
6 
6 


8112847 
8184620 
8256242 
8327714 
8399037 




003164557 
003154574 
003144654 
003134796 
003125000 


321 
322 
323 
324 
325 


103041 
103684 
104329 
104976 
105625 


33076161 
33386248 
33698267 
34012224 
34328125 


17.9164729 
17.9443584 
17.9722008 
18.0000000 
18-0277564 


6 
6 
6 
6 
6 


8470213 
8541240 
8612120 
8682855 
8753443 




003115265 
003105590 
003095975 
003086420 
003076923 


326 
327 
328 
329 
330 


106276 
106929 
107584 
108241 
108900 


34645976 
34965783 
35287552 
35611289 
35937000 


18.0554701 
18.0831413 
18.1107703 
18.1383571 
18.1659021 


6 
6 
6 
6 
6 


8823888 
8894188 
8964345 
9034359 
9104232 




003067485 
003058104 
003048780 
003039514 
003030303 


331 
332 
333 
334 
335 


109561 
110224 
110889 
111556 
112225 


36264691 
36594368 
36926037 
37259704 
37595375 


18.1934054 
18.2208672 
18.2482876 
18.2756669 
18.3030052 


6 
6 
6 
6 
6 


9173964 
9243556 
9313008 
9382321 
9451496 




003021148 
003012048 
003003003 
002994012 
002985075 


336 
337 
338 
339 
340 


112896 
113569 
114244 
114921 
115600 


37933056 
38272753 
38614472 
38958219 
39304000 


18.3303028 
18.3575598 
18.3847763 
18.4119526 
18.4390889 


6 
6 
6 
6 
6 


9520533 
8589434 
9658198 
9726826 
9795321 




002976190 
002967359 
002958580 
002949853 
002941176 


341 
342 
343 
344 
345 


116281 
116964 
117649 
118336 
119025 


39651821 
40001688 
40353607 
40707584 
41063625 


18.4661853 
18.4932420 
18.5202592 
18.5472370 
18.5741756 


6 
6 
7 
7 
7 


9863681 
9931906 
0000000 
0067962 
0135791 




002932551 
002923977 
002915452 
002906977 
002898551 


346 
347 
348 
349 
350 


119716 
120409 
121104 
121801 
122500 


41421736 
41781923 
42144192 
42508549 
42875000 


18.6010752 
18.6279360 
18.6547581 
18.6815417 
18.7082869 


7 
7 
7 
7 
7 


0203490 
0271058 
0338497 
0405806 
0472987 




002890173 
002881844 
002873563 
002865330 
002857143 


351 
352 
353 
354 
355 


123201 
123904 
124609 
125316 
126025 


43243551 
43614208 
43986977 
44361864 
44738875 


18.7349940 
18.7616630 
18.7882942 
18-8148877 
18.8414437 


7 
7 
7 
7 
7 


0540041 
0606967 
0673767 
0740440 
0806988 




002849003 
002840909 
002832861 
002824859 
002816901 


356 
357 
358 
359 
360 


126736 
127449 
128164 
128881 
129600 


45118016 
45499293 
45882712 
46268279 
46656000 


18.8679623 
18.8944436 
18.9208879 
18.9472953 
18.9736660 


7 
7 
7 
7 
7 


0873411 
0939709 
1005885 
1071937 
1137866 




002808989 
002801120 
002793296 
002785515 
002777778 



739 



TABLE XXXII.—SQUARES, CUBES. 


SQUARE ROOTS, 

- 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


361 
362 
363 
364 
365 


130321 
131044 
131769 
132496 
133225 


47045881 
47437928 
47832147 
48228544 
48627125 


19-0000000 
19-0262976 
19-0525589 
19-0787840 
19-1049732 


7.1203674 
7-1269360 
7.1334925 
7-1400370 
7.1465695 




002770083 
002762431 
002754821 
002747253 
002739726 . 


366 
367 
368 
369 
370 


133956 
134689 
135424 
136161 
136900 


49027896 
49430863 
49836032 
50243409 
50653000 


19.1311265 
19-1572441 
19.1833261 
19.2093727 
19-2353841 
19-2613603 
19.2873015 
19.3132079 
19.3390796 
19.3649167 


7-1530901 
7-1595988 
7-1660957 
7.1725809 
7.1790544 




002732240 
002724796 
002717391 
002710027 
002702703 . 


371 
372 
373 
374 
375 


137641 
138384 
139129 
139876 
140625 


51064811 
51478848 
51895117 
52313624 
52734375 


7-1855162 
7-1919663 
7-1984050 
7-2048322 
V-2112479 




002695418 : 

002688172 

002680965 

002673797 

002666667 


376 
377 
378 
379 
380 


141376 
142129 
142884 
143641 
144400 


53157376 
53582633 
54010152 
54439939 
54872000 


19.3907194 
19.4164878 
19.4422221 
19.4679223 
19-4935887 


7-2176522 
7-2240450 
7-2304268 
7-2367972 
7.2431565 




002659574 
002652520 
002645503 
002638522 
002631579 


381 
382 
383 
384 
385 


145161 
145924 
146689 
147456 
148225 


55306341 
55742968 
56181887 
56623104 
57066625 


19-5192213 1 7-2495045 
19-5448203 ' 7-2558415 
19-5703858 7-2621675 
19.5959179 7-2684824 
19.6214169 1 7-2747864 




002624672 
002617801 
002610966 
002604167 
002597403 


386 
387 
388 
389 
390 


148996 
149769 
150544 
151321 
152100 


57512456 19.6468827 | 7-2810794 
57960603 19-6723156 ■ 7-2873617 
58411072 19-6977156 I 7-2936330 
58863869 19-7230829 1 7-2998936 
59319000 19.7484177 7-3061436 




002590674 
002583979 
002577320 
002570694 
002564103 


391 
392 
393 
394 
395 


152881 
153664 
154449 
155236 
156025 


59776471 19-7737199 
60236288 19-7989899 
60698457 i 19-8242276 
61162984 : 19-8494332 
61629875 19-8746069 


7-3123828 
7.3186114 
7.3248295 
7.3310369 
7.3372339 




002557545 
002551020 
002544529 
002538071 
002531646 


396 
397 
398 
399 
400 


156816 
157609 
158404 
159201 
160000 


62099136 19-8997487 
62570773 19-9248588 
63044792 ' 19-9499373 
63521199 1 19-9749844 
64000000 20.0000000 


7.3434205 
7-3495966 
7-3557624 
7-3619178 
7-3680630 




002525253 
002518892 
002512563 
002506268 
002500000 


401 
402 
403 
404 
405 


160801 
161604 
162409 
163216 
164025 


64481201 
64964808 
65450827 
65939264 
66430125 


20-0249844 7-3741979 
20-0499377 7-3803227 
20-0748599 7-3864373 
20-0997512- i 7-3925418 
20-1246118 7.3986363 




002493766 
002487562 
002481390 
002475248 
002469136 


406 
407 
408 
409 
410 


164836 
165649 
166464 
167281 
168100 


66923416 
67419143 
67917312 
68417929 
68921000 


20-1494417 7-4047206 
20-1742410 7-4107950 
20-1990099 7.4168595 
20.2237484 7-4229142 
20.248.4567 7.4289589 




002463054 
002457002 
002450980 
002444988 
002439024 


411 
412 
413 
414 
415 


168921 
169744 
170569 
171396 
172225 


69426531 
69934528 
70444997 
70957944 
71473375 


20-2731349 

20-2977831 

20-3224014 

, 20-3469899 

' 20-3715488 


7-4349938 
7.4410189 
7-4470342 
7-4530399 
7.4590359 




002433090 
002427184 
002421308 
002415459 
002409639 


416 
417 
418 
419 
420 


173056 
173889 
174724 
175561 
176400 


71991296 
72511713 
73034632 
73560059 
74088000 


20-3960781 
20-4205779 
20-4450483 
20-4694895 
20-4939015 


7-4650223 
7.4709991 
7-4769664 
7.4829242 
7.4888724 




002403846 
002398082 
002392344 
002386635 
00238^35$^ 



740 







CUBE ROOTS 


AND RECIPROCALS. 




No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


121 
422 
423 
424 
425 


177241 
178084 
178929 
179776 
180625 


74618461 
75151448 
75686967 
76225024 
76765625 


20.5182845 
20-5426386 
20-5669638 
20-5912603 
20-6155281 


7. 
7. 
7- 
7- 
7. 


4948113 
5007406 
5066607 
5125715 
5184730 


.002375297 
.002369668 
•002364066 
.002358491 
002352941 


426 
427 
428 
429 
430 


181476 
182329 
183184 
184041 
184900 


77308776 
77854483 
78402752 
78953589 
79507000 


20-6397674 
20-6639783 
20-6881609 
20-7123152 
20. 7364414 


7 
7 
7 
7 
7 


5243652 
5302482 
5361221 
5419867 
5478423 


.002347418 
.002341920 
.002336449 
.002331002 
•002325581 


431 
432 
433 
434 
435 


185761 
186624 
187489 
188356 
189225 


80062991 
80621568 
81182737 
81746504 
82312875 


20-7605395 
20-7846097 
20-8086520 
20-8326667 
20.8566536 


7 
7 
7 
7 
7 


5536888 
5595263 
5653548 
5711743 
5769849 


.002320186 
.002314815 
.002309469 
.002304147 
.002298851 


436 
437 
438 
439 
440 


190096 
190969 
191844 
192721 
193600 


82881856 
83453453 
84027672 
84604519 
85184000 


20.8806130 
20.9045450 
20-9284495 
20.9523268 
20.9761770 


7 
7 
7 
7 
7 


5827865 
5885793 
5943633 
6001385 
6059049 


.002293578 
.002288330 
.002283105 
.002277904 
.002272727 


441 
442 
443 
444 
445 


194481 
195364 
196249 
197136 
198025 


85766121 
86350888 
86938307 
87528384 
88121125 


21.0000000 
21.0237960 
21-0475652 
21-0713075 
21-0950231 


7 
7 
7 
7 
7 


6116626 
-6174116 
.6231519 
-6288837 

6346067 


•002267574 
.002262443 
.002257336 
.002252252 
•002247191 


446 
447 
448 
449 
450 


198916 
199809 
200704 
201601 
202500 


88716536 
89314623 
89915392 
90518849 
91125000 


21-1187121 
21-1423745 
21-1660105 
21.1896201 
21-2132034 


7 
7 
7 
7 
7 


-6403213 
-6460272 
.6517247 
-6574138 
-6630943 


.002242152 
.002237136 
.002232143 
.002227171 
•002222222 


451 
452 
453 
454 
455 


203401 
204304 
205209 
206116 
207025 


91733851 
92345408 
92959677 
93576664 
94196375 


21-2367606 
21.2602916 
21-2837967 
21-3072758 
21-3307290 


7 
7 
7 
7 
7 


-6687665 
-6744303 
-6800857 
6857328 
■6913717 


.002217295 
.002212389 
.002207506 
.002202643 
•002197802 


456 
457 
458 
459 
460 


207936 
208849 
209764 
210681 
211600 


94818816 
95443993 
96071912 
96702579 
97336000 


21-3541565 
21.3775583 
21.4009346 
21.4242853 
21-4476106 


7 
7 
7 
7 

7 


-6970023 
•7026246 
■7082388 
-7138448 
.7194426 


.002192982 
.002188184 
.002183406 
.002178649 
.002173913 


461 
462 
463 
464 
465 


212521 
213444 
214369 
215296 
216225 


97972181 
98611128 
99252847 
99897344 
100544625 


21-4709106 
21-4941853 
21.5174348 
21-5406592 
21-5638587 


7 
7 
7 
7 
7 


•7250325 
•7306141 
•7361877 
•7417532 
-7473109 


.002169197 
.002164502 
.002159827 
.002155172 
.002150538 


466 
467 
468 
469 
470 


217156 
218089 
219024 
219961 
220900 


101194696 
101847563 
102503232 
103161709 
103823000 


21-5870331 
21.6101828 
21-6333077 
21-6564078 
21-6794834 


7 
7 
7 
7 
7 


-7528606 
-7584023 
.7639361 
.7694620 
-7749801 


.002145923 
.002141328 
.002136752 
.002132196 
•002127660 


471 
472 
473 
474 
475 


221841 
222784 
223729 
224676 
225625 


104487111 
105154048 
105823817 
106496424 
107171875 


21.7025344 
21-7255610 
21-7485632 
21-7715411 
21. 7944947 


7 
7 
7 
7 
7 


.7804904 
.7859928 
.7914875 
.7969745 
.8024538 


.002123142 
.002118644 
.002114165 
.002109705 
•002105263 


476 
477 
478 
479 
480 


226576 
227529 
228484 
229441 
230400 


107850176 
108531333 
109215352 
109902239 
110592000 


21-8174242 
21. 8403297 
21.8632111 
21.8860686 
21.9089023 


7 
7 
7 
7 
7 


.8079254 
.8133892 
.8188456 
•8242942 
.8297353 


.002100840 
.002096436 
.002092050 
.002087683 
.002083333 


741 



TABLE XXXII.— SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes, 


Square Roots, 


Cube Roots. 


Reciprocals. 


481 

482 
483 
484 
485 


231361 

232324 
233289 
234256 
235225 


111284641 
111980168 
112678587 
113379904 
114084125 


21.9317122 
21.9544984 
21.9772610 
22-0000000 
22.0227155 


7-8351688 
7-8405949 
7.8460134 
7.8514244 
7-8568281 




.002079002 
.002074689 
.002070393 
-002066116 
.002061856 


486 
487 
488 
489 
490 


236196 
237169 
238144 
239121 
240100 


1 114791256 

1 115501303 

i 116214272 

116930169 

117649000 


22-0454077 
22-0680765 
22.0907220 
22.1133444 
22.1359436 


7-8622242 
7-8676130 
7-8729944 
7-8783684 
7.8837352 




-002057613 
.002053388 
.002049180 
-002044990 
-002040816 


491 
492 
493 
494 
495 


241081 
242064 
243049 
244036 
245025 


118370771 
119095488 
119823157 
120553784 
121287375 


22-1585198 
22.1810730 
22.2036033 
22.2261108 
22-2485955 


7-8890946 
7.8944468 
7-8997917 
7-9051294 
7.9104599 




-002036660 
-002032520 
-002028398 
-002024291 
.002020202 


496 
497 
498 
499 
500 


246016 
247009 
248004 
249001 
250000 


122023936 
122763473 
123505992 
124251499 
125000000 


22.2710575 
22-2934968 
22-3159136 
22.3383079 
22.3606798 


7-9157832 
7-9210994 
7.9264085 
7.9317104 
7.9370053 




.002016129 
-002012072 
-002008032 
•002004008 
-002000000 


501 
502 
503 
504 
505 


251001 
252004 
253009 
254016 
255025 


125751501 
126506008 
127263527 
128024064 
128787625 


22-3830293 
22-4053565 
22-4276615 
22-4499443 
22.4722051 


7.9422931 
7-9475739 
7-9528477 
7-9581144 
7-9633743 




001996008 
001992032 
001988072 
001984127 
001980198 


506 
507 
508 
509 
510 


256036 
257049 
258064 
259081 
260100 


129554216 , 

130323843 

131096512 

131872229 

132651000 


22-4944438 
22-5166605 
22-5388553 
22.5610283 
22-5831796 


7-9686271 
7.9738731 
7-9791122 
7.9843444 
7.9895697 




001976285 
001972387 
001968504 
001964637 
001960784 


511 
512 
513 
514 
515 


261121 
262144 
263169 
264196 
265225 


133432831 
134217728 
135005697 
135796744 
136590875 


22-6053091 
22-6274170 
22.6495033 
22-6715681 
22.6936114 


7.9947883 
8-0000000 
8-0052049 
8-0104032 
8.0155946 




001956947 
001953125 
001949318 
001945525 
001941748 


516 
517 
518 
519 
520 


266256 
267289 
268324 
269361 
270400 


137388096 
138188413 
138991832 
139798359 
140608000 


22-7156334 
22-7376340 
22-7596134 
22.7815715 
22.8035085 


8.0207794 
8-0259574 
8-0311287 
8-0362935 
8.0414515 




001937984 
001934236 
001930502 
001926782 
001923077 


521 
522 
523 
524 
525 


271441 
272484 
273529 
274576 
276625 


141420761 
142236648 
143055667 
1438f7824 
144703125 


22.8254244 
22.8473193 
22.8691933 
22.8910463 
22.9128785 


8-0466030 
8-0517479 
8.0568862 
8.0620180 
8.0671432 




001919386 
001915709 
001912046 
001908397 
001904762 


526 
527 
528 
529 
530 


276676 
277729 
278784 
279841 
280900 


145531576 
146363183 
147197952 
148035889 
148877000 


22.9346899 
22.9564806 
22-9782506 
23-0000000 
23.0217289 


8.0722620 
8.0773743 
8.0824800 
8-0875794 
8-0926723 




001901141 
001897533 
001893939 
001890359 
001886792 


531 
532 
533 
534 
535 


281961 
283024 
284089 
285156 
286225 


149721291 
150568768 
151419437 
152273304 
153130375 


23-0434372 
23-0651252 
23.0867928 
23.1084400 
23.1300670 


8-0977589 
8-1028390 
8-1079128 
8-1129803 
8-1180414 




001883239 
001879699 
001876173 
001872659 
001869159 


536 
537 
538 
539 
540 


287296 
288369 
289444 
290521 
291600 


153990656 
154854153 
155720872 
156590819 
157464000 


23-1516738 
23-1732605 
23-1948270 
23-2163735 
23-2379001 


8-1230962 
8-1281447 
8-1331870 
8-1382230 
8-1432529 




00186567i 
001862197 
001858736 
001855288 
001S51852 



742 







CUBE ROOTS 


, AND RECIPROCALS. 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


541 
542 
543 
544 
545 


292681 
293764 
294849 
295936 
297025 


158340421 
159220088 
160103007 
160989184 
161878625 


23.2594067 
23-2808935 
23.3023604 
23.3238076 
23-3452351 


8 
8 
8 
8 
8 


1482765 
1532939 
1583051 
1633102 
1683092 


.001848429 
-001845018 
-001841621 
-001838235 
-001834862 


546 
547 
548 
549 
550 


298116 
299209 
300304 
301401 
302500 


162771336 
163667323 
164566592 
165469149 
166375000 


23.3666429 
23.3880311 
23.4093998 
23-4307490 
23.4520788 


8 
8 
8 
8 
8 


1733020 
1782888 
1832695 
1882441 
1932127 


.001831502 
-001828154 
.001824818 
.001821494 
-001818182 


551 
552 
553 
554 
555 


303601 
304704 
305809 
306916 
308025 


167284151 
168196608 
169112377 
170031464 
170953875 


23-4733892 
23.4946802 
23-5159520 
23-5372046 
23- 5584380 


8 
8 
8 
8 
8 


1981753 
2031319 
2080825 
2130271 
2179657 


.001814882 
.001811594 
-001808318 
.001805054 
-001801802 


556 
557 
558 
559 
560 


309136' 
310249 
311364 
312481 
313600 


171879616 
172808693 
173741112 
174676879 
175616000 


23.5796522 
23-6008474 
23.6220236 
23.6431808 
23.6643191 


8 
8 
8 
8 
8 


2228985 
2278254 
.2327463 
2376614 
2425706 


.001798561 
-001795332 
-001792115 
.001788909 
-001785714 


561 
562 
563 
564 
565 


314721 
315844 
316969 
318096 
319225 


176558481 
177504328 
178453547 
179406144 
180362125 


23.6854386 
23-7065392 
23-7276210 
23.7486842 
23-7697286 


8 
8 
8 
8 
8 


2474740 
2523715 
2572633 
2621492 
2670294 


-001782531 
-001779359 
-001776199 
.001773050 
.001769912 


566 
567 
568 
569 
570 


320356 
321489 
322624 
323761 
324900 


181321496 
182284263 
183250432 
184220009 
185193000 


23-7907545 
23.8117618 
23.8327506 
23-8537209 
23. 8746728 


8 
8 
8 
8 
8 


2719039 
2767726 
2816355 
2864928 
2913444 


-001766784 
-001763668 
-001760563 
001757469 
-001754386 


571 
572 
573 
574 
575 


326041 
327184 
328329 
329476 
330625 


186169411 
187149248 
188132517 
189119224 
190109375 


23-8956063 
23.9165215 
23-9374184 
23-9582971 
23.9791576 


8 
8 
8 
8 
8 


2961903 
3010304 
3058651 
3106941 
3155175 


.001751313 
-001748252 
-001745201 
.001742160 
-001739130 


576 
577 
578 
579 
580 


331776 
332929 
334084 
335241 
336400 


191102976 
192100033 
193100552 
194104539 
195112000 


24-0000000 
24.0208243 
24.0416306 
24-0624188 
24.0831891 


8 
8 
8 
8 
8 


3203353 
3251475 
3299542 
3347553 
3395509 


-001736111 
.001733102 
.001730104 
-001727116 
-001724138 


581 
582 
583 
584 
585 


337561 
338724 
339889 
341056 
342225 


196122941 
197137368 
198155287 
199176704 
200201625 


24-1039416 
24-1246762 
24-1453929 
24-1660919 
24.1867732 


8 
8 
8 
8 
8 


3443410 
3491256 
3539047 
3586784 
3634466 


-001721170 
-001718213 
-001715266 
-001712329 
-001709402 


586 
587 
588 
589 
590 


343396 
344569 
345744 
346921 
348100 


201230056 
202262003 
203297472 
204336469 
205379000 


24-2074369 
24-2280829 
24-2487113 
24-2693222 
24-2899156 


8 
8 
8 
8 
8 


3682095 
3729668 
3777188 
3824653 
3872065 


-001706485 
-001703578 
-001700680 
.001697793 
-001694915 


591 
592 
593 
594 
595 


349281 
350464 
351649 
352836 
354025 


206425071 
207474688 
208527857 
209584584 
210644875 


24-3104916 
24-3310501 
24 3515913 
24-3721152 
24-3926218 


8 
8 
8 
8 
8 


3919423 
3966729 
4013981 
4061180 
4108326 


.001692047 
-001689189 
-001686341 
-001683502 
-001680672 


596 
597 
598 
599 
600 


355216 
356409 
357604 
358801 
360000 


211708736 
212776173 
213847192 
214921799 
216000000 


24-4131112 
24-4335834 
24-4540385 
24-4744765 
24-4948974 


8 
8 
8 
8 
8 


4155419 
4202460 
4249448 
4296383 
4343267 


-001677852 
.001675042 
.001672241 
.001669449 
.001666667 



743 



TABLE XXXII.— SQUARES, CUBES 


SQUARE ROOTS, i 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


601 
602 
603 
604 
605 


361201 
362404 
363609 
364816 
366025 


217081801 
218167208 
219256227 
220348864 
221445125 


24.5153013 
24.5356883 
24.5560583 
24.5764115 
24.5967478 


8 
8 
8 
8 
8 


.4390098 
.4436877 
-4483605 
.4530281 
-4576906 


.001663894 
.001661130 
.001658375 
.001655629 
.001652893 


606 
607 
608 
609 
610 


367236 
368449 
369664 
370881 
372100 


222545016 
223648543 
224755712 
225866529 
226981000 


24.6170673 
24.6373700 
24.6576560 
24-6779254 
24-6981781 


8 
8 
8 
8 
8 


-4623479 
-4670001 
-4716471 
-4762892 
4809261 


.001650165 
.001647446 
.001644737 
.001642036 
.001639344 


611 
612 
613 
614 
615 


373321 
374544 
375769 
376996 
378225 


228099131 
229220928 
230346397 
231475544 
232608375 


24.7184142 
24-7386338 
24.7588368 
24-7790234 
24-7991935 


8 
8 
8 
8 
8 


-4855579 
-4901848 
-4948065 
-4994233 
.5040350 


.001636661 
.001633987 
.001631321 
.001628664 
.001626016 


616 
617 
618 
619 
620 


379456 
380689 
381924 
383161 
384400 


233744896 
234885113 
236029032 
237176659 
238328000 


24-8193473 
24-8394847 
24-8596058 
24-8797106 
24-8997992 


8 
8 
8 
8 
8 


.5086417 

-5132435 

-5178403 

-5224321 

5270189 


.001623377 
.001620746 
.001618123 
.001615509 
.001612903 


621 
622 
623 
624 
625 


385641 
386884 
388129 
389376 
390625 


239483061 
240641848 
241804367 
242970624 
244140625 


24-9198716 
24-9399278 
24-9599679 
24.9799920 
25.0000000 


8 
8 
8 
8 
8 


5316009 

5361780 

-5407501 

-5453173 

5498797 


.001610306 
.001607717 
.001605136 
.001602564 
.001600000 


626 
627 
628 
629 
630 


391876 
393129 
394384 
395641 
396900 


245314376 
246491883 
247673152 
248858189 
250047000 


25.0199920 
25.0399681 
25.0599282 
25.0798724 
25-0998008 


8 
8 
8 
8 
8 


5544372 
5589899 
5635377 
5680807 
5726189 


.001597444 
.001594896 
.001592357 
.001589825 
.001587302 


631 
632 
633 
634 
635 


398161 
399424 
400689 
401956 
403225 


251239591 
252435968 
253636137 
254840104 
256047875 


25-1197134 
25-1396102 
25.1594913 
25.1793566 
25.1992063 


8 
8 
8 
8 
8 


5771523 
5816809 
5862047 
5907238 
5952380 


.001584786 
.001582278 
.001579779 
.001577287 
.001574803 


636 
637 
638 
639 
640 


404496 
405769 
407044 
408321 
409600 


257259456 
258474853 
259694072 
260917119 
262144000 


25.2190404 
25.2388589 
25.2586619 
25.2784493 
25-2982213 


I 

8 
8 
8 


5997476 
6042525 
6087526 
6132480 
6177388 


.001572327 
.001569859 
001567398 
.001564945 
•001562500 


641 
642 
643 
644 
645 


410881 
412164 
413449 
414736 
416025 


263374721 
264609288 
265847707 
267089984 
268336125 


25.3179778 
25.3377189 
25.3574447 
25.3771551 
25.3968502 


8 
8 
8 
8 
8 


6222248 
6267063 
6311830 
6356551 
6401226 


.001560062 
.001557632 
.001555210 
.001552795 
.001550388 


646 
647 
648 
649 
650 


417316 
418609 
419904 
421201 
422500 


269586136 
270840023 
272097792 
273359449 
274625000 


25.4165301 
25.4361947 
25.4558441 
25.4754784 
25-4950976 


8 
8 
8 
8 
8 


6445855 
6490437 
6534974 
6579465 
6623911 


.001547988 
.001545595 
.001543210 
.001540832 
.001538462 


651 
652 
653 
654 
655 


423801 
425104 
426409 
427716 
429025 


275894451 
277167808 
278445077 
279726264 
281011375 


25-5147016 
25.5342907 
25.5538647 
25.5734237 
25.5929678 


8 
8 
8 
8. 
8 


6668310 
6712665 
6756974 
6801237 
6845456 


.001536098 
.001533742 
.001531394 
.001529052 
.0J1526718 


656 
657 
658 
659 
660 


430336 
431649 
432964 
434281 
435600 


282300416 
283593393 
284890312 
286191179 
287496000 


25.6124969 
25.6320112 
25.6515107 
25.6709953 
25.6904652 


8. 
8. 
8. 
8. 
8. 


6889630 
6933759 
6977843 
7021882 
7065877 


.001524390 
.001522070 
.001519757 
.001517451 
.001515152 



744 



CUBE ROOTS. AND RECIPROCALS. 


No. 


Square?. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


661 
662 
663 
664 
665 


436921 
438244 
439569 
440896 

442225 


288804781 
290117528 
291434247 
292754944 
294079625 


25.7099203 
25.7293607 
25.7487864 
25.7681975 
25.7875939 


8 
8 
8 
8 
8 


7109827 
7153734 
7197596 
7241414 
7285187 


.001512859 
.001510574 
.001508296 
.001506024 
.001503759 


666 
667 
668 
669 
670 


443556 
444889 
446224 
447561 
448900 


295408296 
296740963 
298077632 
299418309 
300763000 


25.8069758 
25.8263431 
25.8456960 
25.8650343 
25.8843582 


8 
8 
8 
8 
8 


7328918 
7372604 
7416246 
7459848 
7503401 


.001501502 
.001499250 
.001497006 
.001494768 
.001492537 


671 
672 
673 
674 
675 


450241 
451584 
452929 
454276 
455625 


302111711 
303464448 
304821217 
306182024 
307546875 


25.9036677 
25.9229628 
25-9422435 
25.9615100 
25-9807621 


8 
8 
8 
8 
8 


7546913 
.7590383 
7633809 
7677192 
7720532 


.001490313 
.001488095 
.001485884 
.001483680 
-001481481 


676 
677 
678 
679 
680 


456976 
458329 
459684 
461041 
462400 


308915776 
310288733 
311665752 
313046839 
314432000 


26-0000000 
26.0192237 
26.0384331 
26.0576284 
26-0768096 


8 
8 
8 
8 
8 


7763830 
7807084 
7850296 
7893466 
7936593 


.001479290 
.001477105 
.001474926 
.001472754 
.001470588 


681 
682 
683 
684 
685 


463761 
465124 
466489 
467856 
469225 


315821241 
317214568 
318611987 
320013504 
321419125 


26.0959767 
26.1151297 
26.1342687 
26.1533937 
26-17250^7 


8 
8 
8 
8 
8 


7979679 
8022721 
8065722 
8108681 
8151598 


.001468429 
.001466276 
.001464129 
.001461988 
-001459854 


686 
687 
688 
689 
690 


470596 
471969 
473344 
474721 
476100 


322828856 
324242703 
325660672 
327082769 
328509000 


26.19160_7 
26.2106848 
26.22975 1 
26.2488095 
26.2678511 


8 
8 
8 
8 
8 


8194474 
8237307 
8280099 
8322850 
8365559 


.001457726 
.001455604 
.001453488 
.001451379 
.001449275 


691 
692 
693 
694 
695 


477481 
478864 
480249 
481636 
483025 


329939371 
331373888 
332812557 
334255384 
335702375 


26.286° 89 
26.3058.29 
26.3248932 
26.34G8797 
26-3628527 


8 
8 
8 
8 
8 


8408227 
8450854 
8493440 
8535985 
8578489 


.001447178 
.001445087 
.001443001 
.001440922 
.001438849 


696 
697 
698 
699 
700 


484416 
485809 
487204 
488601 
490000 


337153536 
338608873 
340068392 
341532099 
343000000 


26.3818119 
26.4007576 
26.4196896 
26.4386081 
26-4575131 


8 
8 
8 
8 
8 


8620952 
8663375 
8705757 
8748099 
8790400 


.001436782 
•001434720 
.001432665 
.001430615 
■001428571 


701 
702 
703 
704 
705 


491401 
492804 
494209 
495616 
497025 


344472101 
345948408 
347428927 
348913664 
350402625 


26-4764046 
26-4952826 
26-5141472 
26.5329983 
26-5518361 


8 
8 
8 
8 
8 


8832661 
8874882 
8917063 
8959204 
9001304 


.001426534 
-001424501 
.001422475 
.001420455 
-001418440 


706 
707 
708 
709 
710 


498436 
499849 
501264 
502681 
504100 


351895816 
353393243 
354894912 
356400829 
357911000 


26-5706605 
26-5894716 
26-6082694 
26-6270539 
26-6458252 


8 
8 
8 
8 
8 


9043366 
9085387 
9127369 
9169311 
9211214 


.001416431 
.001414427 
.001412429 
.001410437 
.001408451 


711 
712 
713 
714 
715 


505521 
506944 
508369 
509796 
511225 


359425431 
360944128 
362467097 
363994344 
365525875 


26.6645833 
26.6833281 
26.7020598 
26-7207784 
26-7394839 


8 
8 
8 
8 
8 


9253078 
9294902 
9336687 
9378433 
9420140 


.001406470 
.001404494 
.001402525 
.001400560 
.001398S01 


716 
717 
718 
719 
720 


512656 
514089 
515524 
516961 
518400 


367061696 
368601813 
370146232 
371694959 
373248000 


26.7581763 
26.7768557 
26.7955220 
26.8141754 
26.8328157 


8 
8 
8 
8 
8 


9461809 
9503438 
9545029 
9586581 
9628095 


.001396648 
.001394700 
.001392758 
.001390821 
.001388889 



745 



TABLE XXXII.—SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


721 
722 
723 
724 
725 


519841 
521284 
522729 
524176 
525625 


374805361 
376367048 
377933067 
379503424 
381078125 


26 
26 
26 
26 
26 


8514432 
8700577 
•8886593 
•9072481 
•9258240 


8 
8 
8 
8 
8 


.9669570 
.9711007 
■9752406 
•9793766 
•9835089 




001386963 
001385042 
•001383126 
.001381215 
.001379310 


726 
727 
728 
729 
730 


527076 
528529 
529984 
531441 
532900 


382657176 
384240583 
385828352 
387420489 
389017000 


26 
26 
26 
27 
27 


•9443872 
•9629375 
•9814751 
•0000000 
•0185122 


8 
8 
8 

9 
9 


.9876373 
.9917620 
9958829 
.0000000 
•0041134 




.001377410 
.001375516 
.001373626 
.001371742 
.001369863 


731 
732 
733 
734 
735 


534361 
535824 
537289 
538756 
540225 


390617891 
392223168 
393832837 
395446904 
397065375 


27 
27 
27 
27 
27 


•0370117 
•0554985 
•0739727 
•0924344 
•1108834 


9 
9 
9 
9 
9 


•0082229 
•0123288 
•0164309 
•0205293 
•0246239 




.001367989 
.001366120 
.001364256 
.001362398 
.001360544 


736 
737 
738 
739 
740 


541696 
543169 
544644 
546121 
547600 


398688256 
400315553 
401947272 
403583419 
405224000 


27 
27 
27 
27 
27 


-1293199 
•1477439 
•1661554 
•1845544 
•2029410 


9 
9 
9 
9 
9 


.0287149 
.0328021 
0368857 
•0409655 
•0450419 




.001358696 
.001356852 
.001355014 
•001353180 
001351351 


741 
742 
743 
744 
745 


549081 
550564 
552049 
553536 
555025 


406869021 27 
408518488 27 
410172407 27 
411830784 27 
413493625 27 


•2213152 
•2396769 
•2580263 
•2763634 
2946881 


9 
9 
9 
9 
9 


•0491142 
•0531831 
•0572482 
•0613098 
0653677 




.001349528 
.001347709 
.001345895 
.001344086 
001342282 


746 
747 
748 
749 
750 


556516 
558009 
559504 
561001 
562500 


415160936 
416832723 
418508992 
420189749 
421875000 


27 
27 
27 
27 

27 


3130006 
3313007 
3495887 
3678644 
3861279 


9 
9 
9 
9 
9 


0694220 
0734726 
0775197 
0815631 
0856030 




001340483 
001338688 
001336898 
001335113 
001333333 


751 
752 
753 
754 
755 


564001 
565504 
567009 
568516 
570025 


423564751 
425259008 
426957777 
428661064 
430368875 


27 
27 
27 
27 
27 


4043792 
4226184 
4408455 
4590604 
4772633 


9 
9 
9 
9 
9 


0896392 
0936719 
0977010 
1017265 
1057485 




001331558 
001329787 
001328021 
001326260 
001324503 


756 
757 
758 
759 
760 


571536 
573049 
574564 
576081 
577600 


432081216 
433798093 
435519512 
437245479 
438976000 


27 
27 
27 
27 
27 


4954542 
5136330 
5317998 
5499546 
5680975 


9 
9 
9 
9 
9 


1097669 
1137818 
1177931 
1218010 
1258053 




001322751 
001321004 
001319261 
001317523 
001315789 


761 
762 
763 
764 
765 


579121 
580644 
582169 
583696 
585225 


440711081 
442450728 
444194947 
445943744 
447697125 


27 
27 
27 
27 
27 


5862284 
6043475 
6224546 
6405499 
6586334 


9 
9 
9 
9 
9 


1298061 
1338034 
1377971 
1417874 
1457742 




001314060 
001312336 
001310616 
001308901 
001307190 


766 
767 
768 
769 
770 


586756 449455096 
588289 451217663 
589824 452984832 
591361 ! 454756609 
592900 i 456533000 


27 
27 
27 
27 
27 


6767050 
6947648 
7128129 
7308492 
7488739 


9 
9 

I 

9 


1497576 
1537375 
1577139 
1616869 
1656565 




001305483 
001303781 
001302083 
001300390 
001298701 


771 
772 
773 
774 
775 


594441 1 458314011 
595984 1 460099648 
597529 461889917 
599076 463684824 
600625 465484375 


27 
27 
27 
27 
27 


7668868 
7848880 
8028775 
8208555 
8388218 


9 
9 

9 


1696225 
1735852 
1775445 
1815003 
1854527 




001297017 
001295337 
001293661 
001291990 
001290323 


776 
777 
778 
779 
780 


602176 
603729 
605284 
606841 
608400 


467288576 
469097433 
470910952 
472729139 
474552000 


27 
27 
27 
27. 
27. 


8567766 
8747197 
8926514 
9105715 
9284801 


9^ 
9. 
9. 
9. 
9. 


1894018 
1933474 
1972897 
2012286 
2051641 




001288660 
001287001 
001285347 
001283697 
001282051 



746 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots, 


Reciprocals. 


781 
782 
783 
784 
785 


609961 
611524 
613089 
614656 
616225 


476379541 
478211768 
480048687 
481890304 
483736625 


27 
27 
27 
28 
28 


9463772 
9642629 
9821372 
0000000 
0178515 


9 
9 
9 
9 
9 


2090962 
2130250 
2169505 
2208726 
2247914 




001280410 
001278772 
001277139 
001275510 
001273885 


786 
787 
788 
789 
790 


617796 
619369 
620944 
622521 
624100 


485587656 
487443403 
489303872 
491169069 
493039000 


28 
28 
28 
28 
28 


0356915 
0535203 
0713377 
0891438 
1069386 


9 
9 
9 
9 
9 


2287068 
2326189 
2365277 
2404333 
2443355 




001272265 
001270648 
001269036 
001267427 
001265823 


791 
792 
793 
794 
795 


625681 
627264 
628849 
630436 
632025 


494913671 
496793088 
498677257 
500566184 
502459875 


28 
28 
28 
28 
28 


1247222 
1424946 
1602557 
1780056 
1957444 


9 
9 
9 
9 
9 


2482344 
2521300 
2560224 
2599114 
2637973 




001264223 
001262626 
001261034 
001259446 
001257862 


796 
797 
798 
799 
800 


633616 
635209 
636804 
638401 
640000 


504358336 
506261573 
508169592 
510082399 
512000000 


28 
28 
28 
28 
28 


2134720 
2311884 
2488938 
2665881 
2842712 


9 
9 
9 
9 
9 


2676798 
2715592 
2754352 
2793081 
2831777 




001256281 
001254705 
001253133 
001251564 
001250000 


801 
802 
803 
804 
805 


641601 
643204 
644809 
646416 
648025 


513922401 
515849608 
517781627 
519718464 
521660125 


28 
28 
28 
28 
28 


3019434 
3196045 
3372546 
3548938 
3725219 


9 
9 
9 
9 
9 


2870440 
2909072 
2947671 
2986239 
3024775 




001248439 
001246883 
001245330 
001243781 
001242236 


806 
807 
808 
809 
810 


649636 
651249 
652864 
654481 
656100 


523606616 
525557943 
527514112 
529475129 
531441000 


28 
28 
28 
28 
28 


3901391 
4077454 
4253408 
4429253 
4604989 


9 
9 
9 
9 
9 


3063278 
3101750 
3140190 
3178599 
3216975 




001240695 
001239157 
001237624 
001236094 
001234568 


811 
812 
813 
814 
815 


657721 
659344 
660969 
662596 
664225 


533411731 
535387328 
537367797 
539353144 
541343375 


28 
28 
28 
28 
28 


4780617 
4956137 
5131549 
5306852 
5482048 


9 
9 
9 
9 
9 


3255320 
3293634 
3331916 
3370167 
3408386 




001233046 
001231527 
001230012 
001228501 
001226994 


816 
817 
818 
819 

820 


665856 
667489 
669124 
670761 
672400 


543338496 
545338513 
547343432 
549353259 
551368000 


28 
28 
28 
28 
9.8 


5657137 
5832119 
6006993 
6181760 
6356421 


9 
9 
9 
9 
9 


3446575 
3484731 
3522857 
3560952 
3599016 




001225490 
001223990 
001222494 
001221001 
001219512 


821 
822 
823 
824 
825 


674041 
675684 
677329 
678976 
680625 


553387661 
555412248 
557441767 
559476224 
561515625 


28 
28 
28 
28 
28 


6530976 
6705424 
6879766 
7054002 
7228132 


9 
9 
9 
9 
9 


3637049 
3675051 
3713022 
3750963 
3788873 




001218027 
001216545 
001215067 
001213592 
001212121 


826 
827 
828 
829 
830 


682276 
683929 
685584 
687241 
688900 


563559976 
565609283 
567663552 
569722789 
571787000 


28 
28 
28 
28 
28 


7402157 
7576077 
7749891 
7923601 
8097206 


9 
9 
9 
9 
9 


3826752 
3864600 
3902419 
3940206 
3977964 




001210654 
001209190 
001207729 
001206273 
001204819 


831 
832 
833 
834 
835 


690561 
692224 
693889 
695556 
697225 
698896 
700569 
702244 
703921 
705600 


573856191 
575930368 
578009537 
580093704 
582182875 
584277056 
586376253 
588480472 
590589719 
592704000 


28 
28 
28 
28 
28 


8270706 
8444102 
8617394 
8790582 
8963666 


9 
9 
9 
9 
9 


4015691 
4053387 
4091054 
4128690 
4166297 




001203369 
001201923 
001200480 
001199041 
001197605 


836 
837 
838 
839 
840 


28 
28 
28 
28 
28 


9136646 
9309523 
9482297 
9654967 
9827535 


9 
9 
9 
9 
9 


4203873 
4241420 
4278936 
4316423 
4353880 




001196172 
001194743 
001193317 
001191895 
001190476 



747 



TABLE XXXII.— SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. I 


leciprocals. 


841 


707281 


594823321 


29.0000000 


9.4391307 


001189061 


842 


708934 


596947688 


29 


0172363 


9 


4428704 


001187648 


843 


710649 


599077107 


29 


0344623 


9 


4466072 


001186240 


844 


712336 


601211584 


29 


0516781 


9 


4503410 


001184834 


845 


714025 


603351125 


29 


0688837 


9 


4540719 


001183432 


846 


715716 


605495736 


29 


0860791 


9 


4577999 


001182033 


847 


717409 


607645423 


29 


1032644 


9 


4615249 


001180638 


848 


719104 


609800192 


29 


1204396 


9 


4652470 


001179245 


849 


720801 


611960049 


29 


1376046 


9 


4689661 


.001177856 


850 


722500 


614125000 


29 


1547595 


9 


4726824 


.001176471 


851 


724201 


616295051 


29 


1719043 


9 


4763957 


.001175088 


852 


725904 


618470208 


29 


1890390 


9 


4801061 


.001173709 


853 


727609 


620650477 


29 


2061637 


9 


4838136 


.001172333 


854 


729316 


622835864 


29 


2232784 


9 


4875182 


.001170960 


855 


731025 


625026375 


29 


2403830 


9 


4912200 


.001169591 


856 


732736 


627222016 


29 


2574777 


9 


4949188 


.001168224 


857 


734449 


629422793 


29 


2745623 


9 


4986147 


.001166861 


858 


736164 


631628712 


29 


2916370 


9 


5023078 


.001165501 


859 


737881 


633839779 


29 


3087018 


9 


5059980 


.001164144 


860 


739600 


,636056000 


29 


3257566 


9 


5096854 


•001162791 


861 


741321 


638277381 


29 


3428015 


9 


5133699 


.001161440 


862 


743044 


640503928 


29 


3598365 


9 


5170515 


.001160093 


863 


744769 


642735647 


29 


3768616 


9 


5207303 


.001158749 


864 


746496 


644972544 


29 


3938769 


9 


5244063 


.001157407 


865 


748225 


647214625 


29 


4108823 


9 


5280794 


001156069 


866 


749956 


649461896 


29 


4278779 


9 


5317497 


.001154734 


867 


751689 


651714363 


9 


4448637 


9 


5354172 


001153403 


868 


753424 


653972032 


29 


4618397 


9 


5390818 


•001152074 


869 


755161 


656234909 


29 


4788059 


9 


5427437 


.001150748 


870 


756900 


658503000 


29 


4957624 


9 


5464027 


001149425 


871 


758641 


660776311 


29 


5127091 


9 


5500589 


.001148106 


872 


760384 


663054848 


29 


5296461 


9 


5537123 


.001146789 


873 


762129 


665338617 


29 


5465734 


9 


5573630 


.001145475 


874 


763876 


667627624 


29 


5634910 


9 


5610108 


-001144165 


875 


765625 


669921875 


29 


5803989 


9 


5646559 


.001142857 


876 


767376 


672221376 


29 


5972972 


9 


5682982 


^001141553 


877 


769129 


674526133 


29 


6141858 


9 


5719377 


001140251 


878 


770884 


676836152 


29 


6310648 


9 


5755745 


001138952 


879 


772641 


679151439 


29 


6479342 


9 


5792085 


001137656 


880 


774400 


681472000 


29 


6647939 


9 


5828397 


001136364 


881 


776161 


683797841 


29 


6816442 


9 


5864682 


001135074 


882 


777924 


686128968 


29 


6984848 


9 


5900939 


001133787 


883 


779689 


688465387 


29 


7153159 


9 


5937169 


001132503 


884 


781456 


690807104 


29 


7321375 


9 


5973373 


001131222 


885 


783225 


693154125 


29 


7489496 


9 


6009548 


001129944 


886 


784996 


695506456 


29 


7657521 


9 


6045696 


001128668 


887 


786769 


697864103 


29 


7825452 


9 


6081817 


001127396 


888 


788544 


700227072 


29 


7993289 


9 


6117911 


001126126 


889 


790321 


702595369 


29 


8161030 


9 


6153977 


001124859 


890 


792100 


704969000 


29 


8328678 


9 


6190017 


001123596 


891 


793881 


707347971 


29 


8496231 


9 


6226030 


001122334 


892 


795664 


709732288 


29 


8663690 


9 


6262016 


001121076 


893 


797449 


712121957 


29 


8831056 


9 


6297975 


001119821 


894 


799236 


714516984 


29 


8998328 


9 


6333907 


001118568 


895 


801025 


716917375 


29 


9165506 


9 


6369812 


001117318 


896 


802816 


719323136 


29 


9332591 


9 


6405690 


001116071 


897 


804609 


721734273 


29 


9499583 


9 


6441542 


001114827 


898 


806404 


724150792 


29 


9666481 


9 


6477367 


001113586 


899 


808201 


726572699 


29 


9833287 


9 


6513166 


001112347 


900 


810000 


729000000 


30.0000000 


9.6548938 


001111111 



748 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots, 


Cube Roots. 


Reciprocals. 


901 
902 
903 
904 
905 


811801 
813604 
815409 
817216 
819025 


731432701 
733870808 
736314327 
738763264 
741217625 


30 
30 
30 
30 
30 


0166620 
0333148 
0499584 
0665928 
0832179 


9 
9 
9 
9 
9 


6584684 
6620403 
6656096 
6691762 
6727403 




001109878 
001108647 
001107420 
001106195 
001104972 


906 
907 
908 
909 
910 


820836 
822649 
824464 
826281 
828100 


743677416 
746142643 
748613312 
751089429 
753571000 


30 
30 
30 
30 
30 


0998339 
1164407 
1330383 
1496269 
1662063 


9 
9 
9 
9 
9 


6763017 
6798604 
6834166 
6869701 
6 05211 




001103753 
001102536 
001101322 
001100110 
001098901 


911 
912 
913 
914 
915 


829921 
831744 
833569 
835396 
837225 


756058031 
758550528 
761048497 
763551944 
766060875 


30 
30 
30 
30 
30 


1827765 
1993377 
2158899 
2324329 
2489669 


9 
9 
9 
9 
9 


6940694 
6976151 
7011583 
7046989 
7082369 




001097695 
001096491 
001095290 
001094092 
001092896 


916 
917 
918 
919 
920 


839056 
840889 
842724 
844561 
846400 


768575296 
771095213 
773620632 
776151559 
778688000 


30 
30 
30 
30 
30 


2654919 
2820079 
2985148 
3150128 
3315018 


9 
9 
9 
9 
9 


7117723 
7153051 
7188354 
7223631 
7258883 




001091703 
001090513 
001089325 
001088139 
001086957 


921 
922 
923 
924 
925 


848241 
850084 
851929 
853776 
855625 


781229961 
783777448 
786330467 
788889024 
791453125 


30 
30 
30 
30 
30 


3479818 
3644529 
3809151 
3973683 
4138127 


9 
9 
9 
9 
9 


7294109 
7329309 
7364484 
7399634 
7434758 




001085776 
001084599 
001083423 
001082251 
001081081 


926 
927 
928 
929 
930 


857476 
859329 
861184 
863041 
864900 


794022776 
796597983 
799178752 
801765089 
804357000 


30 
30 
30 
30 
30 


4302481 
4466747 
4630924 
4795013 
4959014 


9 
9 
9 
9 
9 


7469857 
7504930 
7539979 
7575002 
7610001 




001079914 
001078749 
001077586 
001076426 
001075269 


931 
932 
933 
934 
935 


866761 
868624 
870489 
872356 
874225 


806954491 
809557568 
812166237 
814780504 
817400375 


30 
30 
30 
30 
30 


5122926 
5286750 
5450487 
5614136 
5777697 


9 
9 
9 
9 
9 


7644974 
7679922 
7714845 
7749743 
7784616 




001074114 
001072961 
001071811 
001070664 
001069519 


936 
937 
938 
939 
940 


876096 
877969 
879844 
881721 
883600 


820025856 
822656953 
825293672 
827936019 
830584000 


30 
30 
30 
30 
30 


5941171 
6104557 
6267857 
6431069 
6594194 


9 
9 
9 
9 
9 


7819466 
7854288 
7889087 
7923861 
7958611 




001068376 
001067236 
001066098 
001064963 
001063830 


941 
942 
943 
944 
945 


885481 
887364 
889249 
891136 
893025 


833237621 
835896888 
838561807 
841232384 
843908625 


30 
30 
30 
30 
30 


6757233 
6920185 
7083051 
7245830 
7408523 


9 
9 
9 
9 
9 


7993336 
8028036 
8062711 
8097362 
8131989 




001062699 
001061571 
001060445 
001059322 
001058201 


946 
947 
948 
949 
950 


894916 
896809 
898704 
900601 
902500 


846590536 
849278123 
851971392 
854670349 
857375000 


30 
30 
30 
30 
30 


7571130 
7733651 
7896086 
8058436 
8220700 


9 
9 
9 
9 
9 


8166591 
8201169 
8235723 
8270252 
8304757 




001057082 
001055966 
001054852 
001053741 
001052632 


951 
952 
953 
954 
955 


904401 
906304 
908209 
910116 
912025 


860085351 
862801408 
865523177 
868250664 
870983875 


30 
30 
30 
30 
30 


8382879 
8544972 
8706981 
8868904 
9030743 


9 
9 
9 
9 
9 


8339238 
8373695 
8408127 
8442536 
8476920 




001051525 
001050420 
001049318 
001048218 
001047120 


956 
957 
958 
959 
960 


913936 
915849 
917764 
919681 
921600 


873722816 
876467493 
879217912 
881974079 
884736000 


30 
30 
30 
30 
30 


9192497 
9354166 
9515751 
9677251 
9838668 


9 
9 
9 
9 
9 


8511280 
8545617 
8579929 
8614218 
8648483 




001046025 
001044932 
001043841 
001042753 
001041667 



749 



TABLE XXXII.— SQUARES, CUBES, ETC. 



Squares. 



923521 
925444 
927369 
929296 
931225 



933156 
935089 
937024 
938961 
940900 



942841 
944784 
946729 
948676 
950625 



952576 
954529 
956484 
958441 
960400 



Cubes. 



887503681 
890277128 
893056347 
895841344 
898632125 



901428696 
904231063 
907039232 
909853209 
912673000 



915498611 
918330048 
921167317 
924010424 
926859375 



929714176 
932574833 
935441352 
938313739 
941192000 



Square Roots. 



31.0000000 
31.0161248 
31-0322413 
31.0483494 
31.0644491 



31-0805405 
31-0966236 
31-1126984 
31.1287648 
31-1448230 



31-1608729 
31-1769145 
31-1929479 
31-2089731 
31.2249900 



31.2409987 
31.2569992 
31-2729915 
31-2889757 
31-3049517 



Cube Roots. 



9-8682724 
9-8716941 
9-8751135 
9-8785305 
9-8819451 



9-8853574 
9-8887673 
9-8921749 
9-8955801 
9-8989830 



9.9023835 
9-9057817 
9.9091776 
9-9125712 
9-9159624 



9-9193513 
9-9227379 
9-9261222 
9-9295042 
9-9328839 



Reciprocals. 



.001040583 
.001039501 
.001038422 
.001037344 
-001036269 



-001035197 
-001034126 
.001033058 
.001031992 
001030928 



.001029866 
.001028807 
.001027749 
.001026694 
-001025641 



.001024590 
.001023541 
.001022495 
-001021450 
-001020408 



962361 
964324 
966289 
968256 
970225 



944076141 
946966168 
949862087 
952763904 
955671625 



31-3209195 
31-3368792 
31-3528308 
31-3687743 
31-3847097 



9-9362613 
9-9396363 
9-9430092 
99463797 
9- 9497479 



972196 
974169 
976144 
978121 
980100 



958585256 
961504803 
964430272 
967361669 
970299000 



31-4006369 
31-4165561 
31.4324673 
31.4483704 
31. 4642654 



9. 9531138 
9.9564775 
9.9598389 
9. 9631981 
9. 9665549 



982081 
984064 
986049 
988036 
990025 



973242271 
976191488 
979146657 
982107784 
985074875 



992016 
994009 
996004 
998001 
1000000 



988047936 
991026973 
994011992 
997002999 
1000000000 



31.4801525 
31.4960315 
31-5119025 
31-5277655 
31-5436206 



9-9699095 
9-9732619 
9-9766120 
9.9799599 
9-9833055 



31-5594677 
31-5753068 
31-5911380 
31. 6069613 
31-6227766 



9-9866488 
9-9899900 
9-9933289 
9-9966656 
10 0000000 



-001019368 
-001018330 
-001017294 
-001016260 
.001015228 



-001014199 
-001013171 
-001012146 
-001011122 
.001010101 



-001009082 
-001008065 
-001007049 
-001006036 
■001005025 



.001004016 
.001003009 
-001002004 
-001001001 
.001000000 



1002001 
1004004 
1006009 
1008016 
1010025 



1006 


1012036 


1007 


1014049 


1008 


1016064 


1009 


1018081 


1010 


1020100 


1011 


1022121 


1012 


1024144 


1013 


1026169 


1014 


1028196 


1015 


1030225 



1032256 
1034289 
1036324 
1038361 
1040400 



1003003001 
1006012008 
1009027027 
1012048064 
1015075125 



31. 6385840 
31.6543836 
31- 6701752 
31. 6859590 
31-7017349 



10.0033322 
10. 0066622 
10.0099899 
10.0133155 
10. 0166389 



1018108216 
1021147343 
1024192512 
1027243729 
1030301000 



31.7175030 
31.7332633 
31-7490157 
31.7647603 
31-7804972 



10-0199601 
10-0232791 
10-0265958 
10-0299104 
100332228 



1033364331 
1036433728 
1039509197 
1042590744 
1045678375 



31.7962262 
31.8119474 
31.8276609 
31.8433666 
31.8590646 



10-0365330 
10-0398410 
10-0431469 
10-0464506 
10-0497521 



1048772096 
1051871913 
1054977832 
1058089859 
1061208000 



31-8747549 
31-8904374 
31.9061123 
31-9217794 
31-9374388 

~75Q 



10.0530514 
10.0563485 
10.0596435 
10.0629364 
10.0662271 



-0009990010 
-0009980040 
-0009970090 
-0009960159 
.0009950249 



-0009940358 
-0009930487 
-0009920635 
-0009910803 
-0009900990 



.0009891197 
.0009881423 
.0009871668 
.0009881933 
-0009852217 



.0009842520 
.0009832842 
.0009823183 
.0009813543 
.0009803922 



TABLE XXXIII.— 


CUBIC YARDS PER 100 FEET OF LEVEL 






SECTIONS. SLOPE 1 : 1 








Depth, 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


d 


12 feet. 


14 feet. 


16 feet. 


18 feet. 


20 feet. 


28 feet. 


30 feet. 


32 feet. 


1 


48 


56 


63 


70 


78 


107 


115 


122 


2 


104 


119 


133 


148 


163 


222 


237 


252 


3 


167 


189 


211 


233 


256 


344 


367 


389 


4 


237 


267 


296 


326 


356 


474 


504 


533 


5 


315 


352 


389 


426 


463 


611 


648 


685 


6 


400 


444 


489 


533 


578 


756 


800 


844 


7 


493 


544 


596 


648 


700 


907 


959 


1011 


8 


593 


652 


711 


770 


830 


1067 


1126 


1185 


9 


700 


767 


833 


900 


967 


1233 


1300 


1367 


10 


815 


889 


963 


1037 


1111 


1407 


1481 


1556 


11 


937 


1019 


1100 


1181 


1263 


1589 


1670 


1752 


12 


1067 


1156 


1244 


1333 


1422 


1778 


1867 


1956 


13 


1204 


1300 


1396 


1493 


1589 


1974 


2070 


2167 


14 


1348 


1452 


1556 


1659 


1763 


2178 


2281 


2385 


15 


1500 


1611 


1722 


1833 


1944 


2389 


2500 


2611 


16 


1659 


1778 


1896 


2015 


2133 


2607 


2726 


2844 


17 


1826 


1952 


2078 


2204 


2330 


2833 


2959 


3085 


18 


2000 


2133 


2267 


2400 


2533 


3067 


3200 


3333 


19 


2181 


2322 


2463 


2604 


2744 


3307 


3448 


3589 


20 


2370 


2519 


2667 


2815 


2963 


3556 


3704 


3852 


21 


2567 


2722 


2878 


3033 


3189 


3811 


3967 


4122 


22 


2770 


2933 


3096 


3259 


3422 


4074 


4237 


4400 


23 


2981 


3152 


3322 


3493 


3663 


4344 


4515 


4685 


24 


3200 


3378 


3556 


3733 


3911 


4622 


4800 


4978 


25 


3426 


3611 


3796 


3981 


4167 


4907 


5093 


5278 


26 


3659 


3852 


4044 


4237 


4430 


5200 


5393 


5585 


27 


3900 


4100 


4300 


4500 


4700 


5500 


5700 


5900 


28 


4148 


4356 


4563 


4770 


4978 


5807 


6015 


6222 


29 


4404 


4619 


4833 


5048 


5263 


6122 


6337 


6552 


SO 


4667 


4889 


5111 


5333 


5556 


6444 


6667 


6889 


81 


4937 


5167 


5396 


5626 


5856 


6774 


7004 


7233 


32 


5215 


5452 


5689 


5926 


6163 


7111 


7348 


7585 


33 


5500 


5744 


5989 


6233 


6478 


7456 


7700 


7944 


34 


5793 


6044 


6296 


6548 


6800 


7807 


8059 


8311 


35 


6093 


6352 


6611 


6870 


7130 


8167 


8426 


8685 


36 


6400 


6667 


6933 


7200 


7467 


8533 


8800 


9067 


37 


6715 


6989 


7263 


7537 


7811 


8907 


9181 


9456 


38 


7037 


7319 


7600 


7881 


8163 


9289 


9570 


9852 


39 


7367 


7656 


7944 


8233 


8522 


9678 


9967 


10256 


40 


7704 


8000 


8296 


8593 


8889 


10074 


10370 


10667 


41 


8048 


8352 


8656 


8959 


9263 


10478 


10781 


11085 


42 


8400 


8711 


9022 


9333 


9644 


10889 


11200 


11511 


43 


8759 


9078 


9396 


9715 


10033 


11307 


11626 


11944 


44 


9126 


9452 


9778 


10104 


10430 


11733 


12059 


12385 


45 


9500 


9833 


10167 


10500 


10833 


12167 


12500 


12833 


46 


9881 


10222 


10563 


10904 


11244 


12607 


12948 


13289 


47 


10270 


10619 


10967 


11315 


11663 


13056 


13404 


13752 


48 


10667 


11022 


11378 


11733 


12089 


13511 


13867 


14222 


49 


11070 


11433 


11796 


12159 


12522 


13974 


14337 


14700 


50 


11481 


11852 


12222 


12593 


12963 


14444 


14815 


15185 


51 


11900 


12278 


12656 


13033 


13411 


14922 


15300 


15678 


52 


12326 


12711 


13096 


13481 


13867 


15407 


15793 


16178 


53 


12759 


13152 


13544 


13937 


14330 


15900 


16293 


16685 


54 


13200 


13600 


14000 


14400 


14800 


16400 


16800 


17200 


55 


13648 


14056 


14463 


14870 


15278 


16907 


17315 


17722 


56 


14104 


14519 


14933 


15348 


15763 


17422 


17837 


18252 


57 


14567 


14989 


15411 


15833 


16256 


17944 


18367 


18789 


58 


15037 


15467 


15896 


16326 


16756 


18474 


18904 


19333 


59 


15515 


15952 


16389 


16826 


17263 


19011 


19448 


19885 


60 


16000 


16444 


16889 


17333 


17778 


19556 


20000 


20444 



761 



TABLE XXXIII.— CUBIC YARDS PER 100 FEET OF LEVEL 
SECTIONS. SLOPE 1.5 : 1. 



Depth 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


d 


12 feet. 


14 feet. 


16 feet. 


18 feet. 


20 feet. 


28 feet. 


30 feet. 


32 feet. 


1 


50 


57 


65 


72 


80 


109 


117 


124 


2 


111 


126 


141 


156 


170 


230 


244 


259 


3 


183 


206 


228 


250 


272 


361 


383 


406 


4 


267 


296 


326 


356 


385 


504 


533 


563 


5 


361 


398 


435 


472 


509 


657 


694 


731 


6 


467 


511 


556 


600 


644 


822 


867 


911 


7 


583 


635 


687 


739 


791 


998 


1050 


1102 


8 


711 


770 


830 


889 


948 


1185 


1244 


1304 


9 


850 


917 


983 


1050 


1117 


1383 


1450 


1517 


10 


1000 


1074 


1148 


1222 


1296 


1593 


1667 


1741 


11 


1161 


1243 


1324 


1406 


1487 


1813 


1894 


1976 


12 


1333 


1422 


1511 


1600 


1689 


2044 


2133 


2222 


13 


1517 


1613 


1709 


1806 


1902 


2287 


2383 


2480 


14 


1711 


1815 


1919 


2022 


2126 


2541 


2644 


2748 


15 


1917 


2028 


2139 


2250 


2361 


2806 


2917 


3028 


16 


2133 


2252 


2370 


2489 


2607 


3081 


3200 


3319 


17 


2361 


2487 


2613 


2739 


2865 


3369 


3494 


3620 


18 


2600 


2733 


2867 


3000 


3133 


3667 


3800 


3933 


19 


2850 


2991 


3131 


3272 


3413 


3976 


4117 


4257 


20 


3111 


3259 


3407 


3556 


3704 


4296 


4444 


4593 


21 


3383 


3539 


3694 


3850 


4006 


4628 


4783 


4939 


22 


3667 


3830 


3993 


4156 


4319 


4970 


5133 


5296 


23 


3961 


4131 


4302 


4472 


4642 


5324 


5494 


5665 


24 


4267 


4444 


4622 


4800 


4978 


5689 


5867 


6044 


25 


4583 


4769 


4954 


5139 


5324 


6065 


6250 


6435 


26 


4911 


5104 


5296 


5489 


5681 


6452 


6644 


6837 


27 


5250 


5450 


5650 


5850 


6050 


6850 


7050 


7250 


28 


5600 


5807 


6015 


6222 


6430 


7259 


7467 


7674 


29 


5961 


6176 


6391 


6606 


6820 


7680 


7894 


8109 


SO 


6333 


6556 


6778 


7000 


7222 


8111 


8333 


8556 


31 


6717 


6946 


7176 


7406 


7635 


8554 


8783 


9013 


32 


7111 


7348 


7585 


7822 


8059 


9007 


9244 


9481 


33 


7517 


7761 


8006 


8250 


8494 


9472 


9717 


9961 


34 


7933 


8185 


8437 


8689 


8941 


9948 


10200 


10452 


35 


8361 


8620 


8880 


9139 


9398 


10435 


10694 


10954 


36 


8800 


9067 


9333 


9600 


9867 


10933 


11200 


11467 


37 


9250 


9524 


9798 


10072 


10346 


11443 


11717 


11991 


38 


9711 


9993 


10274 


10556 


10837 


11963 


12244 


12526 


39 


10183 


10472 


10761 


11050 


11339 


12494 


12783 


13072 


40 


10667 


10963 


11259 


11556 


11852 


13037 


13333 


13630 


41 


11161 


11465 


11769 


12072 


12376 


13591 


13894 


14198 


42 


11667 


11978 


12289 


12600 


12911 


14156 


14467 


14778 


43 


12183 


12502 


12820 


13139 


13457 


14731 


15050 


15369 


44 


12711 


13037 


13363 


13689 


14015 


15319 


15644 


15970 


45 


13250 


135£d 


13917 


14250 


14583 


15917 


16250 


16583 


46 


13800 


14141 


14481 


14822 


15163 


16526 


16867 


17207 


47 


14361 


14709 


15057 


15406 


15754 


17146 


17494 


17843 


48 


14933 


15289 


15644 


16000 


16356 


17778 


18133 


18489 


49 


15517 


15880 


16243 


16606 


16969 


18420 


18783 


19146 


50 


16111 


16481 


16852 


17222 


17593 


19074 


19444 


19815 


61 


16717 


17094 


17472 


17850 


18228 


19739 


20117 


20494 


52 


17333 


17719 


18104 


18489 


18874 


20415 


20800 


21185 


53 


17961 


18354 


18746 


19139 


19531 


21102 


21494 


21887 


54 


18600 


19000 


19400 


19800 


20200 


21800 


22200 


22600 


55 


19250 


19657 


20065 


20472 


20880 


22509 


22917 


23324 


56 


19911 


20326 


20741 


21156 


21570 


23230 


23644 


24059 


57 


20583 


21006 


21428 


21850 


22272 


23961 


24383 


24805 


58 


21267 


21696 


22126 


22556 


22985 


24704 


25133 


25563 


59 


21961 


22398 


22835 


23272 


23709 


25457 


25894 


26331 


60 


22667 


23111 


23556 


24000 


24444 


26222 


26667 


27111 



752 



TABLE 



XXXIII.— CORRECTIVE PERCENTAGE FACTORS FOR 
TABLES OF LEVEL SECTIONS. 



To be applied when cross-sections are not level. See § 95 > 
Side slope = 1.5:1 or /? = 33°4r. 



Trans- 
verse 

surface 
slope. 


6=12 feet 
and d== 


6=20 feet 
and d= 


6=30 feet 
and d= 


a° 


Per- 
cent 


10 
feet. 


20 
feet. 


50 
feet. 


10 
feet. 


20 

feet. 


50 

feet. 


10 
feet. 


20 
feet. 


50 
feet. 


5 
10 
15 
20 
SO 


9 
18 
27 
36 
57 


% 
1.9 
8.2 
21 
46 
327 


% 

1.8 

7.7 

20 

44 
324 


r-8 

7.5 
19 
43 
317 


9.0 
23 
51 
358 


% 

1.8 

8.0 

21 

45 
336 


% 

1.8 

7.6 

20 

44 
321 


% 
2.3 
10.0 
26 
57 
400 


% 

2.0 

8.4 

22 

48 
354 


% 

1.8 

7.7 

20 

44 
326 



Sideslope=l:lor/?=45° 



Trans- 
verse 

surface 
slope. 


6=12 feet 
andd= 


6=20 feet 
and d= 


6 = 30 feet 
and d= 


a° 


Per- 
cent 


10 

feet. 


20 
feet. 


50 
feet. 


10 
feet. 


20 

• feet. 


50 
feet. 


10 
feet. 


20 
feet. 


50 
feet. 


5 
10 
15 
20 
30 


9 

18 
27 
36 
57 


0^-9 
3.7 
9.0 

18 

58 


3.4 
8.2 
16 

53 


% 
0.8 
3.2 
7.8 

15 

50 


4.3 
10.3 
20 
67 


% 
0.9 
3.6 
8.7 

17 

56 


% 
0.8 
3.3 
8.0 

16 

51 


% 

1.2 

5.0 

12.1 

24 

78 


% 
0.9 
4.0 
9.5 

19 

61 


% 
0.8 
3.4 
8.2 

16 

53 



753 



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INDEX. 

Numbers refer to sections except where specifically marked pages (p.). 

Abandonment of existing track 457, c. 

Abutments for trestles 142 

Accelerated motion, application of laws to movement of trains 431 

Accidents, danger of, due to curvature 418 

Accuracy of earthwork computations 94 

numerical example 86 

tunnel surveying 163 

Additional business methods of securing (or losing) it 455 

train to handle a given traffic, cost of — Table XXVII. . . .p. 521 

Adhesion of wheels and rails 333, 334 

Adjustments of dumpy level — Appendix pp. 549, 550 

instruments, general principles — Appendix p. 542 

transit — Appendix pp. 643-547 

wye level — Appendix pp. 547-549 

Advance signals, in block signaling 307 

Advantages of re-location of old lines 456 

tie-plates ,...'... 244 

Air-brakes 336, 337 

Air resistance — see Atmospheric resistance. 

Allowance for shrinkage of earthwork 97 

American locomotives, frame 316 

equalizing levers 324 

running gear 323 

system of tunnel excavation 171 

Aneroid barometer, use in reconnoissance leveling 7 

Angle-bars, cost 358, d 

efficiency of 238 

number per mile of track — Table XVII p. 432 

standard 242 

Angle of slope in earthwork 60 

Annual charge against a tie, at 5% interest — Table XXXIV p. 755 

Apprehension of danger, effect on travel 419, c 

ARCH CULVERTS 191-192 

design 191 

example 192 

Area of culverts, computation 178-183 

A. S. C. E. standard rail sections 226 

Assistant engines — see Pusher engines and Pusher grades. 

Atmospheric resistance, train 342 

755 



756 INDEX. 

Atlantic locomotives, running gear , 323 

Austrian system of tunnel excavation I7I 

Automatic air-brakes 337 

signaling, track circuit 310 

Averaging end areas, volume of prismoid computed by 72 

Axle, effect of parallelism 312 

effect of rigid wheels on 311 

radial, possibilities of 313 

size of standard M.C.B 332 

Balance of grades for unequal traffic 452-454 

determination of relative traffic. . . 454 

general principle 452 

theoretical balance . 453 

Balanced grades for one, two, and three engines — Table XXVIII. ... p. 526 

Baldwin Locomotive Works formula for train resistance 350 

BALLAST.— Chap. VII. 

cost 200, 358, a 

cross-sections 198 

laying. 199 

materials 197 

Banjo signals, in block signallinaj 308 

Barometer, reduction of readings to 32^ F. — Table XI p. 729 

use of aneroid m reconnoissance leveling 7 

Barometric elevations — Table XII p. 730 

coefficients for corrections for temperatures and 

humidity— Table XIII p 730 

Beams, strength of stringers considered as 156 

Bearings, compass, use as check on deflections, 16, 17 

in preliminary surveys 11 

Belgian system of tunnel excavation 171 

Belpaire fire-box , 318 

Blasting • 117-123 

use in loosening earth 107, c 

BLOCK SIGNALING.— Chap. XIV. 

Boiler for locomotive 317, 318 

Boiler-power of locomotives, relation to tractive and cylinder power. . 326 
Bolts — see Track bolts. 

Bonds of railroads, security and profits 369 

Borrow-pits, earthwork 89 

Bowls (or pots) as rail supports 201, 223 

Box-cars, size and capacity 328 

Box culverts 188-190 

old-rail 190 

stone 189 

wooden 188 

Bracing for trestles 140, 141 

design 159 

Brakes — see Train-brakes. 

Brake resistances 346 

Bridge joints (rail) i , 240 

Bridjje spirals « ^ 



INDEX. 757 

Bridges and culverts, as affected by changes in alignment 405, 422 

437, 444, 450 

cost of repairs and renewals 387 

Bridges of standard dimensions for small spans 195 

in block signaling 308 

Bridges, trestles, and culverts on railroads, cost 357 

Broken-stone ballast 197 

Burnettizing (chloride-of-zinc process) for preserving timber 213 

Burnt clay ballast 197 

Capital, railroad, classification of 369 

returns on 369, 370 

Caps (trestle), design 158 

Car mileage, nature and cost — Table XX, p. 460-463, and 400 

Cars 328-332 

brake-beams 330 

capacity and size 328 

causes of deterioration, items 13 and 14 406 

cost of renewals and repairs 391 

as affected by changes in alignment . . 406, 
423, 437, 444, 450 

draft-gear 331 

gauge of wheel and form of wheel-tread 332 

stresses in car frames 329 

truck frames 330 

use of metal 330 

wheels, kinetic energy of 347 

Cars and horses, use in earthwork 109, e 

and locomotives, use in earthwork 109, / 

Carts and horses, use in earthwork 109, a 

Cattle guards 193 

passes 194 

Centre of gravity of side-hill sections, earthwork. 92 

Central angle of a curve 21 

Centrifugal force, coimteracted by superelevation of outer rail 41, 42 

of connecting-rod, etc., of locomotive 325 

Chairs as supports for double-headed rails 226 

Chats for ballast 197 

Chemical composition of rails 232 233, a 

purification of water 281 

Chert for ballast 197 

Cinders for ballast 197 

Circular lead rails for switches 262 

Clark's formula for train resistance 350 

Classification of excavated material 124 

Clearance card in permissive block signaling 304 

spaces in locomotives 321 

Clearing and grubbing for railroads, cost 355 

Coal consumption in locomotives 319 

per car-mile 319 

Columbia locomotives, running gear 323 

Compass, use of, in preliminai;y surveys 11 

Competitive traffic 409 ei aeq. 



758 INDEX. 

Competitive rates, equality, regardless of distance 410 

Compensation for curvature 427, 428 

Compensation for curvature rate 428 

reasons. 427 

rules for 428 

Compensators in block signaling 309 

Compound curves 37-40 

modifications of location 39 

nature and use 37 

multiform, used as transition curves 45 

mutual relations of the parts 38 

Compound sections, earthwork 61 

Computation of earthwork 71-95 

approximate, from profiles. 95 

using a slide rule 80 

Conducting transportation, cost of. . . 393-402 

as affected by changes in curvatiu-e 424 

distance 407 

minor grades .... 437 

ruling grades 444 

pusher engines 450 

Coning wheels, effect • 313 

Connecting curve from a curved track to the inside 273 

from a curved track to the outside 272 

from a straight track 271 

Consolidation locomotives, equalizing levers 324 

frame 316 

running gear 323 

Constants, numerical, in common use — Table XXXI p. 733 

Construction of tunnels 169-174 

Contours, obtained by cross-sectioning 12 

Contractors profit, earthwork 115 

Corbels for trestles • • . 144 

Cost of an additional train to handle a given trafl&c — Table XXVII.. . 445 

of ballast 200 

of blasting « 123 

of chemical treatment of timber •••••••••.. 216 

of earthwork 106 et seq. 

of f ramed-timber trestles 150 

of metal ties 222 

ot pile trestles ••• 134 

OF RAILROADS.— Chap. XVII. 

detailed estimate •••• 362 

of rails 236 

of ties 209 

of transportation 364 

of treating wooden ties 216 

of tunneling , 175 

Counterbalancing for locomotives 325 

Crawford's formula for train resistance 350 

Creosoting for preserving timber * 212 

Cross-country routes — reconnoissance 4 



INDEX. 759 

Crossings, one straight, one curved track 278 

two curved tracks 279 

Crossings, two curved tracks, numerical example 279 

two straight tracks 277 

Cross-over between two parallel curved tracks, reversed curve 275, b 

curved tracks, straight connecting 

curve 275, o 

straight tracks 274 

Cross-sectioning, for earthwork computations 68 

for preliminary surveys 12 

irregular sections for earthwork computations 87 

Cross-sections of ballast 198 

of tunnels. . • 164 

Cross-ties — see Ties. 

Crown-bars in locomotive fire-box 318 

Cubic yards per 100 feet of level sections — ^Table XXXIII pp. 752-754 

CULVERTS AND MINOR BRIDGES.— Chap. VI. 

Culverts, arch 191, 192 

area of waterway 178-183 

iron-pipe 186 

old-rail 190 

reinforced-concrete 190, a 

stone box 189 

tile-pipe 187 

wooden box 188 

CURVATURE.— Chap. XXII. 

compensation for 427, 428 

correction for, in earthwork computations 90-93 

danger of accident due to 418 

effect on cost of conducting transportation 424 

of maintenance of equipment 423 

of maintenance of way 422 

operating expenses of a change of 1 ° — Table XXII , 425 

travel 419 

extremes of sharp 429 

general objections 417 

of existing track, determination 35 

proper rate of compensation 428 

Curve, elements of a 1^ 23 

location by deflections 25 

by middle ordinates 29 

by offsets from long chord 30 

by tangential offsets 28 

by two transits 27 

resistance of trains 311, 312, 345 

effect on cost of conducting transportation . . 424 
maintenance of equipment.. 423 

maintenance of way 422 

Curves, elements of 21 

instrumental work in location 26 

limitations in location 34 

method of computing length 20 

modifications of location , 33 

mutual relations of elements 22 



760 INDEX. 

Curves, obstacles to location 32 

simple, method of designation 18 

use and value of other methods of location (not using a transit). . 31 

Cylinder power of locomotives, relation to boiler and tractive power. . . 326 

Deflecting rods for operating block signals 309 

Deflections for a spiral 48 

Degree of a curve 18 

Design of culverts 177 et seq. 

framed trestles 151-159 

bracing 159 

caps and sills 158 

floor stringers 156 

posts 157 

nutlocks 253 

pile trestles 133 

tie-plates 245 

t rack bolts 252 

tunnels 168 

distinctive systems 171 

Development, definition 5 

example, with map 5 

methods of reducing grade 5 

Disadvantages of re-locations of old lines 457 

DISTANCE.— Chap. XXI. 

effect of change on business done 416 

on division of through rates 411 

effect on operating expenses 405- 408 

justification of decrease to save time 415 

relation to rates and expenses 403 

Distant signals in block signaling 306 

Ditches to drain road-bed 64 

Dividends actually paid on railroad stock 369 

Double-ender locomotives, running gear 323 

Double-track, distance between centers 62 

Draft gear Z3l 

"continuous** 331 

Drainage of road-bed, value of 64, 65 

Drains in tunnels 168 

Draw -bars 331 

Drilling holes for blasting 118, 119 

Driving-wheels of locomotives 323 

section of 325 

Drop tests for train resistance 349 

Durability of metal ties 219 

rails 234, 235 

wooden ties 204 

Dynamometer tests of train resistance 348 

Earnings of railroads, estimation of 373 

per mile of road 373 

EARTHWORK.— Chap. III. 

Earthwork computations, accuracy 94 

approximate computations from profiles... 95 

probable error 85 



INDEX. 761 

Earthwork computations, relation of actual volume to numerical result 66 

Earthwork, cost 106 et seq., 356 

limit of free haul 105 

method of computing haul 99 et seq. 

shrinkage 96 

surveys 66-70 

Eccentricity of center of gravity of earthwork cross-section 91 

Economics, railroad, justification of methods of computation 367,426 

nature and limitations 365 

of ties 202 

of treated ties 217 

Elements of a 1° curve 23 

simple curve 21 

transition curves — Table IV pp. 560-568 

Embankments, method of formation 98 

usual form of cross-section 58 

Empirical formulae for culvert area , 180 

accuracy required 183 

value 181 

Engineering, proportionate and actual cost, in railroad construction. . 353 

Engineering News formula for pile-driving 131 

for train resistance 350 

Engineer's duties in locating a railroad 366 

Engine-houses for locomotives 289 

Enginemen, basis of wages 394, 407 

English system of tunnel excavation 171 

Enlargement of tunnel headings 170 

Entrained water in steam 321 

Equalizing-levers on locomotives 324 

Equipment, effect of curvature on maintenance of 423 

Equivalent level sections in earthwork, determination of ar«a 78 

sections in earthwork, determination of area 77 

Estimation of probable volume of traffic and of probable growth 373 

Excavation, usual form of cross-section 58 

Exhaust-steam, effect of back-pressure 321 

Expansion of rails 230 

Explosives, amount used 120 

firing 122 

tamping 121 

use in blasting 117 

Expenditure of money for railroad purposes, general principles 377 

External distance, simple curve 21 

table of, for a 1° curve — Table II p. 556 

Factors of safety, design of timber trestles 155 

Failures of rail joints 241 

Fastenings for metal cross-ties 221 

Field work for locating a simple curve 26 

a spiral 50 

Fire-box of locomotive 318 

required area 318 

Fire-brick arches in locomotive fire-box 318 



762 INDEX. 

Fire protection on trestles 148 

Five-level sections in earthwork, computation of area 81 

Fixed charges, nature and ratio to total disbursements 378 

Flanges of wheels, form 332 

Flanging locomotive driving-wheels, effect 314 

Floor systems for trestling 143-150 

Formation of embankments, earthwork 96-98 

railroad corporations, method 368 

Formulae for pile-driving 131 

required area of culverts 180 

train resistance 350 

trigonometrical — Table XXX pp. 731-732 

useful, and constants — Table XXXI p. 733 

Forney's formula for train resistance 350 

Fouling point of a siding 310 

Foundations for framed trestles 139 

FRAMED TRESTLES 135-159 

abutments , 142 

bracing 140, 141 

cost 150 

design 135, 151-159 

foundations 139 

joints 136 

multiple story construction 137 

span 138 

Frame of locomotive, construction 316 

Free haul of earthwork, limit of 105 

Freight yards 295-299 

general principles 295 

minor yards 297 

relation of yard to main track 296 

track scales 299 

transfer cranes 288 

French system of tunnel excavation 171 

Frictior, laws of, as applied to braking trains 334 

Frof^s, diagrammatic design 255 

Elliot, illustrated in Plate VIII, opposite p. 291. 

for switches 255, 256 

to find frog number 256 

trigonometrical functions — Table III, p. 559. 
Weir, illustrated in Plate VIII, opposite p. 291. 

Fuel for locomotives, cost of 395, 407, item 22, and Table XX 

as affected by changes in alignment, 407, 424, 

437, 444, 450 

German system of tunnel excavation 171 

GRADE.— Chap. XXIII. 

(see Minor grades, Pusher grades. Ruling grades.) 

accelerated motion ot trains on 431 

distinction between ruling and minor grades 430 

in tunnels 165 

line, change in, based on mass diagram 104 

resistance of , 344 



INDEX. 763 

Grade* undulatory, advantages, disadvantages, and safe limits 434 

virtual 432 

use, value, and misuse 433 

Grade resistance of trains 344 

Gravel ballast 197 

Gravity tests of train resistance 349 

Grate area of locomotives 318, 320 

ratio to total heating surface 320 

Gravity, effect on trains on grades 344 

tests of train resistance 349 

Ground levers for switches 259 

Growth of railroad traffic 373 

affected by increase of facilities 375 

Guard rails for switches 261 

for trestles 145 

Guides around curves and angles (signaling mechanism) 309 

Gun-powder pile-drivers 130 

Hand-brakes 335 

Haul of earthwork, computation of length 99 et seq. 

cost 109, 116, 124, 125 

limit of profitable 116 

method depending on distance hauled 110 

Headings in tunnels 169 

Heating surface in locomotives 320 

Henderson's formula for train resistance 350 

Hire of equipment (rolling-stock), nature of item and cost — ^Table XX. 400 

Hoosac Tunnel, surveys for 160, 163 

I-beam bridges, standard 195 

IMPROVEMENT OF OLD LINES.— Chap. XXIV. 

classification 455 

Inertia resistances 347 

Instrumental ^ork in locating simple curves 26 

spirals 50 

Interest on cost of railroads during construction 360 

Iron pipe culverts 186 

Irregular prismoids, volume 83 

numerical example 84 

sections in earthwork, computation of area 82 

Joints, framed trestles 136 

rail 237-243 

Journal friction of axles 343, b 

Kinetic energy of trains 431 

Kyanizing (bichloride-of-mercury or corrosive-sublimate process) for 

preserving timber 214 

Land and Land damages, cost • 354 

Lateral bracing for trestles «. 141 

Length of rails 229 

a simple curve 20 

a spiral 53 

Level, dumpy, adjustments of — Appendix P. 549 

wye, adjustments of — Appendix P» 547 

Leveling, location surveys 16 



764 INDEX. 

Level sections, volume of prismoids surveyed as 75 

numerical example 76 

Life of locomotives , 327 

Limitations in location of track 34 

of maximum curvature 429 

Lining of tunnels 166 

Loading earthwork, cost 108 

of trestles 154 

Local traffic, definition and distinction from through 409 

Location of stations at distance from business centers, effect 376 

Location Surveys — paper location. 15 

surveying methods 16 

Locomotives, as affected by changes in alignment . . . 406, 423, 437, 444, 450 

causes of deterioration, item 12 406 

cost of renewals and repairs 390 

general structure 316-326 

life of 327 

types permissible on sharp curvature 420, b 

(For details, look for the particular item.) 

Logarithmic sines and tangents of small angles — Table VI p. 589 

sines, cosines, tangents, and cotangents — Table VII. . p. 592 

versed sines and external secants — Table VIII p. 637 

Logarithms of numbers — Table V p. 569 

Long chords for a 1° curve — Table II p. 556 

of a simple curve 21 

Longitudinal bracing of a trestle 140 

Longitudinals (rails) 201, 224 

Loop — see Spiral. 

Loosening earthwork, cost 107 

Loss in traffic due to lack of facilities 376 

Lundie's formula for train resist<ince 350 

Magnitude of railroad business 363 

Maintenance of equipment, as affected by changes in curvature 423 

distance 406 

minor grades 437 

ruling grades 444 

pusher-engines 450 

cost of 389-392 

Maintenance of way as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruhng grades 444 

pusher-engines 450 

cost of 384-388 

Maps, use of, in reconnaissance 6 

Mass curve, area 102 

properties 101 

diagram, effect of change of grade line 104 

haul of earthwork 100 

value 103 

Mathematical design of switches 262-275 

Measurements, location surveys 16 



INDEX. 765 

Mechanism of brakes 335-337 

METAL TIES 218-223 

cost 222 

durability 219 

extent of use 218 

fastenings 221 

form and dimensions 220 

Middle areas, volume of prismoid computed from 73 

ordinate of a simple curve 21 

Mileage, car 400 

locomotives, average annual 327 

MINOR GRADES 435-439 

basis of cost 435 

classification 436 

effect on operating expenses 437 

estimate of cost of one foot of change of elevation. . . . 438 
operating value of the removal of a hump in a 

grade 439 

Minor openings in road-bed 193-195 

Minor stations, rooms required, construction 287 

MISCELLANEOUS STRUCTURES AND BUILDINGS.— Chap. XII. 

Modifications in location, compound curves 39 

simple curves 33 

Mogul locomotives, running gear 323 

Monopoly, extent to which a railroad may be such 371 

Mountain routes — reconnoissance 5 

•* Mud " ballast 197 

sills, trestle foundations 139, 5 

Multiform compound curves used as spirals 45 

Multiple story construction for trestles 137 

Myer*s formula for culvert area 180 

Natural sines, cosines, tangents, and cotangents — Table IX p. 683 

versed sines and external secants — Table X p. 706 

Non-competitive traffic, definition 409 

effect of variations in distance 413, 414 

extent of monopoly 371 

Notes — fonn for cross-sectioning 12 

location surveys 17 

reconnoissance 7 

Number of a frog, to find 256 

of trains per day, probable 374 

Nut-locks, design 253 

N. Y. Central formula for train resistance . . ." 350 

Obstacles to location of trackwork 32 

Obstructed curve, in curve location 32, c 

Old lines, improvement of — Chap. XXIV 

rail culverts 190 

Open cuts vs. tunnels 174 

OPERATING EXPENSES.— Chap. XX. 

detailed classification— Table XX. pp. 460-463 
effect of change of grade — 439, and Table 

XXIV..... p. 511 



766 INDEX. 

operating expenses, effect of curvature on 421-426 

distance on 405-408 

estimated cost of each additional foot — Table 

XXI p. 478 

of each additional mile — Table 

XXI p. 478 

of 1° additional of central angle 

— Table XXII p. 492 

fourfold distribution 379 

per train mile 380 

reasons for uniformity per train mile 381 

(For details look for the particular item.) 

Operation of trains, effect of curvature on 420 

Oscillatory and concussive velocity resistances, train 342 

Ordinates of a spiral 47 

Paper location m location surveys 15 

Passengers carried one mile 363 

Physical tests of steel splice bars 243, a 

steel rails 233, a 

Picks, use in loosening earth 107, b 

Pile bents 129, 133 

driving 130 

driving formulae 131 

points and shoes 132 

trestles, cost 134 

design 133 

PILE TRESTLES 129-134 

Pilot truck of locomotive, action 315 

PIPE CULVERTS 184-187 

advantages 184 

construction 185 

iron 186 

tile 187 

Pipe compensator 309 

Pipes, use in block signaling . 309 

Pit cattle guards 193 

Platforms, station 286 

Ploughs, use in loosening earth 107, a 

Point of curve 21 

inaccessible, in curve location 32, b 

Point of tangency 21 

inaccessible, in curve location 32, b 

Point-rails of switches, construction 258 

Point-switches 258 

Pony truck of locomotive, action 315 

Portals, tunnels, methods of excavation 173 

Posts, trestle, design of 157 

Preliminary financiering of railroads — Chap. XIX., and 352 

Preliminary surveys — cross-section method 11 

•'first " and "second" 14 

general character , 10 



INDEX. 767 

Preliminary surveys, value of re-surveys at critical points 14 

Preservative processes for timber, cost 216 

general principle 210 

methods 211-215. 217 

Prismoidol correction for irregular prismoids, approximate value 84 

in earthwork computations, comparison of exact 

and approximate methods 85, 86 

formula, proof 71 

Prismoid, irregular, computation of volume 83 

Prismoids, in earthwork computations 67 

Profit and loss, dependence on business done 372 

small margin between them for railroad promoters 370 

Profits (and security) in the two general classes of railroad obligations. . . 369 

Profit, in earthwork operations 115 

PROMOTION OF RAILROAD PROJECTS.— Chap. XIX. 

Pumping, for locomotive water-tanks 283, 284 

Pusher grades 446-451 

comparative cost 450, 451 

general principles 446 

required balance between through and pusher grades. . 447 

required length 449 

Pusher engines, cost per mile — Table XXIX p. 529 

operation 448 

service 450 

Radiation from locomotives 321 

into the exhaust-steam 321 

Radii of curves — Table I p. 552 

Radius of curvature (of track), relation to operating expenses 421 

Rail braces 244 

expansion, resistance at joints and ties to free expansion 251 

FASTENINGS.— Chap. X. 

gap, effect of, at joints 239 

joints 237-243, a 

"Bonzano" 243 

•'Cloud'' 243 

"Continuous" 243 

••Fisher" 243 

••Weber" 243 

•*100 per-cent" 243 

••bridge" 240 

effect of rail gap 239 

efficiency of angle-bar 238 

failures 241 

later designs 243 

specifications 243, a 

••supported" 238,240 

•'suspended " 238, 240 

theoretical requirements for perfect 237 

• sections 225, 226 

A.S.C.E ....,,,,, .226 



768 INDEX. 

Rail sections, ''bridge " 225 

"bull-headed" 225, 226 

compound 240 

"pear" section 225 

radius of upper corner, effect 226 

reversible 226 

"Stevens" 225 

"Vignoles" 225 

RAILS.— Chap. IX. 

branding 233-a,ll 

cast-iron 225 

cost 236, 358, c 

of, as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

of renewals of 234 235, 385 

chemical composition 232, 233, a 

effect of stiffness on traction 228 

expansion 230 

stresses caused by prevention of expansion 230 

rules for allowing for 231 

inspection 233-a-12 

length 229 

allowable variation 233, a, 8 

45- and 60-foot rails 229 

No. 2 233-a-13 

physical properties 233 a, 3 

relation of weight, strength, and stiffness 228 

temperature when exposed to sun 231 

testing 233, 233, a 

tons per mile 358, c 

wear on curves 235 

tangents 234 

weight, allowable variation 233-a-7 

for various kinds of traffic 358 

Rates based on distance, reasons 404 

through, method of division of 410 

Receipts (railroad), effect of distance on 409-416 

Reconnoissance over a cross-country route 4 

surveying, leveling methods 7 

surveys 1-9 

character of 1 

cross-country route 4 

distance measurements 8 

mountain route 5 

selection of general route 2 

value of high grade work • 9 

through a river valley 3 

Reduction of barometer reading to 32° F.— Table XI . p. 729 

Reinforced-concrete culverts 190, a 

Renewal of rails, cost of 284, 235, 385 



INDEX. 769 

Renewal of rails, cost of, as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

Renewal of tie* cost of 208, et seq. 386 

as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

regulations governing it 208 

Repairs and renewals of locomotives, cost 390 

as affected by change of dis- 
tance 406 

curvature 423 

minor grades. . 437 

by ruling grades 444 

Repairs of roadway, cost of 384 

as affected by changes in curvature 422 

distance 405 

minor grades. , . . 437 

ruling grades .... 444 

pusher engines 450 

Repairs, wear, depreciation, and interest on cost of plant ; cost for earth- 
work operations 113 

Replacement of a compound curve by a curve with spirals 53 

simple curve by a curve with spirals 51 

Requirements, nut-locks 251 

perfect rail-joint 237 

spikes 247 

track-bolts 251 

Resistances internal to the locomotive 341 

(see Train Resistance.) 

Revenue, gross, distribution of 378 

Roadbed, form of subgrade 63 

width for single and double track 62 

Roadway, cost of repairs of 384 

as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

Roadways, earthwork operations, cost of keeping in order 112 

Rock ballast 197 

Rock cuts, compound sections 61, 62 

Rolling friction of wheels 343, a 

ROLLING STOCK.— Chap. XIV. 

Rotative kinetic energy of wheels of train 347, 431 

Rules for switch-laying 276 

Ruling grades 440-445 



J 



770 INDEX. 

Ruling grades, choice of. 441 

definition 3, 440 

operating value of a reduction in rate of 445 

proportion of traflSc affected by 443 

Run-off for elevated outer rail 43 

Running gear of locomotives, types 323 

Sag in a grade, operating value of filling of 439 

Sand, used for ballast 197 

Scales, track 298 

Scrapers, use in earthwork 109, d 

Screws and bolts, as rail-fastenings 249 

Section-houses, value, construction 288 

Selection of a general route for a railroad 2 

Semaphore boards, in block signaling 308 

Setting tie-plates, methods 246 

Shafts, tunnel, design 167 

surveying 161 

Shells and small coal, used as ballast 197 

Shifting centers for locomotive pilot trucks, action 315 

Shoveling (hand) of earthwork, cost 108, a 

(steam) of earthwork, cost 108, b 

Shrinkage of earthwork 96 

allowance 97 

Side-hill work, in earthwork computations 88 

correction for curvature 91 

Signaling, block, "absolute" blocking 304 

automatic 305 

manual systems 302-304 

permissive 304 

Signals, mechanical details 308 

Sills for trestles, design 158 

Simple curves 18-36 

Skidding of wheels on rails 333, 334 

Slag, used for ballast 197 

Slide-rule, in earthwork computations 80 

Slipping of wheels on rails, lateral 312 

longitudinal 311 

Slips, for switchwork 279, a 

Slopes in earthwork, for cut and fill 60, 62 

effect and value of sodding 65 

Slope-stake rod, automatic 70 

Slope-stakes, determination of position 69 

Smith's formula for tram resistance 350 

Sodding slopes, effect and value 65 

Spacing of ties 206 

Span of trestles 138 

Specifications for earthwork 125 

steel rails 233, a 

steel splice-bars 243, a 

wooden ties 207 

Speed of trains, reduction due to curvature 419, a 

relation to superelevation of outer rail 41, 42 

relation to tractive adhesion 334, e 

Spikes 247-250 



INDEX. 771 

Spikes, cost 358, d 

driving 248 

number per mile of track 358, d 

requirements in design 247 

**wooden," for plugging spike-holes 250 

Spirals, bridge and tunnel 5 

(see Transition Curves.) 
Splice-bars — (see Angle-bars). 

Split stringers, caps, and sills 129; 143 

Sprague's formula for train resistance 350 

Spreading earthwork, cost Ill 

Stadia method — for preliminary surveys 13 

Stand pipes, for locomotive water-supply 285 

Starting grade at stations, reduction of 460 

Staybolts for locomotive fire-boxes 318 

Stays, in locomotive fire-box 318 

Steam pile-drivers 130 

Steam-shoveling of earthwork 108, h 

Stiffness of rails, effect on traction 228 

Stocks of railroads, security and profits 369 

Stone box culverts 189 

foundations for framed trestles 139, c 

Straight connecting curve between two parallel curved tracks 275, a 

from a curved main track 273 

frog rails, effect on design of switch 263 

point rails, effect on design of switch 264 

Strength of timber 153 

factors of safety 155 

required elements for trestles 152 

Stringer bridges, standard, steel 195 

Stringers, design 156 

for trestle floors 143 

Stub-switches 257 

Subchord, length 19 

Subgrade, of roadbed, form 63 

Superelevation of the outer rail on curves, L. V. R. R. run-off 43 

on trestles 147 

practical rules 42 

standard on N. Y. N. H. 

& H. R. R 42 

theory 41 

Superintendence J cost in earth operations 114 

of conducting transportation 393 

of maintenance of equipment. . 389 

Supported rail-joints 240 

Surface cattle guards 193, h 

surveys for tunneling 160 

Surveys and engineering expenses for railroads, cost 353 

accuracy 163 

for tunneling 160-163 

with compass 11 

Suspended rail-joints 240 



772 INDEX. 

Swinging pilot truck on locomotive 315 

Switchbacks 5 

Switch construction 254 261 

essential elements 254 

frogs 255, 256 

guard rails 261 

point 258 

stands 259 

stub 257 

tie rods 260 

SWITCHES AND CROSSINGS,. .Chap XI 

Switches, mathematical design 262-276 

comparison of methods 266 

using circular lead rails 262 

using straight frog rails 263 

using straight point rails 264 

using straight frog rails and straight 

point rails 265 

Switching engines, running gear 323 

used in pusher-engine service 448 

Switch leads and distances— Table III p. 559 

laying, practical rules 276 

St ands 259 

TABLES. numbers refer to pages, not sections. 

I. Radii of curves 552-555 

II. Tangents, external distances and Jong chords for a 1° curve, 

556-558 

III. Switch leads and distances 559 

IV. Elements of transition curves 560-568 

V. Logarithms of numbers 569-588 

VI. Logarithmic sines and tangents of small angles 589-591 

VII. Logarithmic sines, cosines, tangents, and cotangents. ,. 592-636 

VIII. Logarithmic versed sines and external secants 637-682 

IX. Natural sines, cosines, tangents, and cotangents 683-705 

X. Natural versed sines and external secants 706-728 

XI. Reduction of barometer reading to 32° F /29 

XII. Barometric elevations 730 

XIII. Coefficients for corrections for temperature and humidity. . 730 

XIV. Capacity of cylindrical water-tanks in United States standard 

gallons of 231 cubic inches 329 

XV. Number of cross-ties per mile 430 

XVI. Tons per mile (with cost) of rails of various weights 431 

XVII. Splice bars for varioua weights of rails. 432 

XVIII. Railroad spikes 433 

XIX. Track bolts, average number in a keg of 200 pounds 433 

XX. Classification of operating expenses of all railroads, .... 460-463 

XXI. Effect on operating expenses of change.s in distance 478 

XXII. Effect on operating expenses of changes in curvature 492 

XXIII. Velocity head of trains 503 

XXIV. Effect on operating expenses of 26.4 feet of rise and fall 511 

XXV. Tractive power of locomotives 516 



INDEX. 773 

TABLES. 

XXVI. Total train resistance per ton on various grades 518 

XXVII. Cost of an additional train to handle a given traffic 521 

XXVIII. Balanced grades for one, two, and three engines 526 

XXIX. Cost per mile of a pusher-engine 529 

XXX. Useful trigonometrical formulae 731, 732 

XXXI. Useful formulae and constants 733 

XXXII. Squares, cubes, squares root, cube roots, and reciprocals. .734-750 

XXXIII. Cubic yards per 100 feet of level sections 752-754 

XXXIV. Annual charge against a tie, at 5% interest 755 

Talbot's formula for culvert area 180 

Tamping for blasting 121 

Tangents for a 1° curve — Table II p. 656 

Tangent distance, simple curve 21 

Tanks, water, for locomotives 282 

capacity of cylindrical tanks 282 

track 284 

Temperature allowances, while laying rails 231 

Ten-wheel locomotives, running gear 323 

Telegraph lines for railroads, cost 361 

TERMINALS.— Chap. XIII. 

inconvenient, resulting loss 376 

justification for great expenditures 376 

Terminal pyramids and wedges, in earthwork 59 

Tests for splice bars 243, a 

for rails 233, a 

to measure the efficiency of brakes 338 

Three-level sections in earthwork, determination of area 79 

numerical example 79 

Throw of a switch 262 

Through traffic, definition 409 

division of receipts between roads 410 

effect of changes in distance on receipts 411 

Tie-plates 244-246 

advantages 244 

elements of design 245 

method of setting 246 

Tie rods, for switches 260 

TIES.— Chap. VIII. 

cost of renewal of 208 et seq. 386 

as affected by changes in curvature 422 

distance. . .^ 405 

minor grades 437 

ruling grades. ..... 444 

pusher-engines 450 

metal 218-223 

cost 222 

durability 219 

extent of use 218 

fastenings 221 

form and dimensions 220 

number per mile of track — Table XV p. 430 

on trestles 146 

wooden 203-217 



774 INDEX. 

Ties, wooden, choice of wood 206 

construction 207 

cost 209, 358, b 

dimensions 205 

durability 204 

economics 202 

quality of timber 207 

spacing 206 

specifications 207 

Tile drains, to drain roadbed 64 

pipe culverts 187 

Timber, choice for trestles 149 

piles 129 

ties 203 

strength of 153 

Tons carried one mile 363 

Topographical maps, use of, in reconnaissance 6 

Track bolts, average number in a keg of 200 pounds 358, d 

cost 358, d 

design 252 

essential requirements 251 

number required per mile 358, d 

circuit for automatic signaling 310 

laying on railroads, cost 358, e 

scales 299 

Tractive power of locomotives, Table XXV, p. 475 and 322 

relation to boiler and cylinder power. . 326 

Traffic, classification of 409 

TRAIN-BRAKES 333-339 

automatic 337 

brake-shoes 339 

general principles 333, 334 

hand-brakes 335 

straight air-brakes 336 

tests for efficiency 338 

Train length limited by curvature 420, a 

load, financial value of increasing 444 

maximum on any grade 442 

loads, methods of increasing 455, b, 458 et aeq. 

RESISTANCE.— Chap. XVI. 

formula? for 350 

total, per ton, on various grades — ^Table 

XXVI p. 518 and 444 

service, cost of, 397 and Table XX. 

as affected by changes in alignment, 407, 424, 437, 

444, 450 
supplies and expenses, cost of, in conducting transportation — 

398 and Table XX. 
wages — (see Train service). 

Transfer cranes in freight yards 298 

Transit, adjustments of pp. 543-547 

Transition curves 41-53 



L 



INDEX. 775 

Transition curves, Table IV PP. 560-568 

application to compound curves 52 

field work - 50 

fundamental principle 44 

replacing a compound curve by curves with spirals 53 
simple curve by a curve with spirals. ... 51 

required length 46 

their relation to tangents and simple curves 49 

to find the deflections from any point 48 

ordinates 47 

use of Table IV 53, a 

Transportation, effect of curvature on conducting 424 

TRESTLES.— Chap. IV. 

cost 150 

extent of use 126 

framed 135-150 

pile 129-134 

posts, design 157 

required elements of strength 152 

sills, design 158 

stringers, design 156 

timber 149, 153 

V8. embankments 127 

Trimming cuts to proper cross-section 112, a 

Trucks, car 330 

four-wheeled, action on curves 312 

locomotive pilot 315 

with shifting center 315 

TUNNELS.— Chap. V. 

cost 175 

vs. open cuts 174 

Tunnel cross-sections 164 

design 164-168 

drains 168 

enlargement 170 

grade 165 

headings 169 

lining 166 

portals 173 

shafts 161, 167 

spirals 5 

Turnout, connecting curve from a straight track 271 

from a curved track to the outside 272 

to the INSIDE 273 

double, from straight track 269 

dimensions, development of approximate rule for above. . . . 267 

from INNER side of curved track 268 

from OUTER side of curved track 267 

Turnouts with straight point rails and straight frog rails, dimensions of 

—Table III p. 559 

two, on same side 270 

Turntables for locomotives 292 

Two-level ground, volume of prismoid surveyed as 74 



776 INDEX. 

Underground surveys in tunnels 162 

Undulatory grades, advantages, disadvantages, and safe limits 434 

Unit chord, simple curves 18 

Upright switch-stands 259 

Useful formulae and constants — Table XXXI p. 733 

trigonometrical formulae — Table XXX p. 731-732 

Valley route — reconnoissance 3 

Velocity head applied to theory of motion of trains 431 

as applied to determination of train resistance 349 

of trains— Table XXIII p. 503 

Velocity of trains, method of obtaining 458 

resistances, train 342 

Ventilation of a tunnel during construction 172 

Vertex inaccessible, curve location 32, o 

of a curve 21 

Vertical curves, mathematical form 56 

necessity for use 54 

numerical example 57 

required length 55 

Virtual grade, reduction of 458-460 

profile, construction of 432 

use, value, and possible misuse 433 

Von Borrie's formula for train resistance 350 

Vulcanizing, for preserving wooden ties 211 

Wages of engine- and roundhouse-men 394; 407, item 21 

as afifected by changes in align- 
ment. .407, 424, 437, 444, 450 

Wagons, use in hauling earthwork 109, b 

Water for locomotives, chemical qualities 281 

consumption and cost 396; 407, item 23 

methods of purification 281 

Stations and water supply 280-285 

location 280 

pumping 283 

required qualities of water 281 

stand-pipes 285 

' tanks 282 

track tanks 284 

table in locomotive fire-box 318 

tanks for locomotives 282 

capacity of cylindrical tanks 282 

protection from freezing 282 

way for culverts 178-183 

Watering stock 369 

Wear of rails on curves 235 

on tangents 234 

Weight of rails, 226, 227, and Table XVT p. 431 

Wellhouse (zinc-tannin) process for preserving timber 215 

Wellington's formula for train resistance 350 

Westinghouse air-brakes 337 

Wheelbarrows, use in hauling earthwork 109, c 

Wheel resistances, train 343 



INDEX. 777 

Wheels and rails, mutual action and reaction 311-315 

effect of rigidly attaching them to axles 311 

White oak, use for trestles 129, 149 

ties 204 

Wire-drawn steam 321 

Wires and pipes, use in block signaling 309 

Wolff's formula for train resistance 350 

Wooden box culverts 188 

spikes, for filling spike holes 250 

YARDS AND TERMINALS— Chap. XIII. 

Yards, engine ' 300 

freight 295-299 

grade in 295 

minor 297 

relation to main tracks 296 

transfer cranes 298 

track scales 299 

Yftlue of proper design 293 



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